Ms M manner W Michigan 5522:» University J w— This is to certify that the dissertation entitled SOME ISOPARAMETRIC HYPERSURFACES IN A COMPLEX HYPERBOLIC SPACE AND THEIR COUNTERPARTS IN ANTI-DE SITTER SPACE TIME presented by Micheal H. Vernon has been accepted towards fulfillment of the requirements for ‘ Ph. D. degreein Mathematics Major professorg Gerald D. Ludden Date 7/ 24/ 85 MS U is an Waive Action/Equal Opportunity Institution 0-12771 “+fl_w ____—~ 4 fi. MSU LlBRARlES .—;—. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ' 937871.38“ : . _ I 5‘”- 'i a :_ ‘ J“) 9-. . o ' I. ll : n 7771777 3 1293 10649 77717, LIE-SEAR? Michimn Qtete University This is to certify that the dissertation entitled SOME ISOPARAMETRIC HYPERSURFACES IN A COMPLEX HYPERBOLIC SPACE AND THEIR COUNTERPARTS IN ANTI-DE SITTER SPACE TIME presented by Micheal H. Vernon has been accepted towards fulfillment of the requirements for ‘ Ph.D. degreein Mathematics Major professori Gerald D. Ludden Date 77124185 MS U i: on ”motive Action/Equal Opportunity Institution 0-12771 I MSU LIBRARIES “ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ' . A931,..33131‘l‘l mi 9 37 SOME ISOPARAMETRIC REAL HYPERSURFACES OF A COMPLEX HYPERBOLIC SPACE AND THEIR COUNT EHPAHT 5 IN ANT I-DE SITTER SPACE TIME 99 Micheal Hugh Vernon A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics l 985 ABSTRACT SOME ISOPARAMET RIC HYPERSURFACES OF A COMPLEX HYPERBOLIC SPACE AND THEIR COUNT ERPARTS IN ANTI-DE SITTER SPACE TIME 39 Micheal Hugh Vernon In this study, real hypersurfaces of a complex hyperbolic space, Le. a complex Riemannian manifold of negative constant holomorphic sectional curvature, that satisfy certain tensor equations are classified by utilizing a Lorentzian hyperbolic Sl-fiber bundle over the ambient complex space. All the hypersurfaces classified are isoparametric (have constant principal curvatures), although this hypothesis is used primarily for congruence. As a byproduct of the classification, some information is gained concerning Sl-invariant hypersurfaces of Lorentzian manifolds of negative constant sectional curvature. The major results are as follows: Theorem 4 A complete, connected contact hypersurface of a complex hyperbolic space of complex dimension n and holomorphic sectional curvature -4 is congruent to one of the following: i) a tube of radius r>0 about a totally geodesic real hyperbolic subspace of dimension n and sectional curvature -l, ii) a tube of radius r>0 about a totally geodesic complex hyperbolic subspace of complex dimension n-i and holomorphic sectional curvature -4, iii) a geodesic hmersphere of radius 00, or iv) ahorosphere. W A complete connected hypersurface of a complex hgperbolic space of complex dimension n and holomorphic sectional curvature -4 whose second fundamental form commutes with the induced almost contact structure is congruent to one of the following: i) a tube of radius r>0 about a totally geodesic complex hyperbolic subspace of complex dimension p, Ospsn-l, and holomorphic sectional curvature -4, or ii) a horosphere. roll A semi-symmetric 5‘-invariant hypersurface of anti-De Sitter space time is congruent to an S'-fiber bundle over one of the hypersurfaces of Theorem 5. I bl.’ O - _‘ ~ {1 \ . I ’I' i‘ g“. .-.,Ilt.‘, .: i 137:7: .1- ,9 .7 \u ‘7‘ " ‘ 5" fl ‘1' no (‘u‘ (TI y)... s O \ ‘I1 1‘ g7 f ' ‘7 'e. \ O .4 \ A 4'. i. i' ) C.) I". \I I“ I I I ' x' \- ,f-_v :fu ) \ v - {Vs i _.1. J "I" ') egwsvp ,, fi. O(\ I.) ”I‘M, f) y’l‘q‘finu‘w‘." ((1‘, I '. IN , v ‘.. EJ‘IT yq.) ':‘. 'IN I C o o . .q‘ ' e .1 t t 4' . b s . I l I a. a W __» I . ' T f f u ‘ , | 0i a ., - DEDICATION This dissertation is dedicated to my loving and beloved wife of eleven years. Without her support none of this would have been possible. ACKNOWLEDGMENTS The author is deeply indebted to Professor Ludden for his patience, guidance and many valuable suggestions in the preparation of this study. I cherish the association I have had with Dr. Ludden and hope to maintain it in years to come. I would like to expess my appreciation to Professors Chen and Blair for their suggestions and sharing their vast knowledge in the subject. Thanks go to Dr. Ralph Howard for many worthwhile conversations concerning tubes in Riemannian manifolds. TABLE OF CONTENTS Introduction, Page i Section 0. Geometric Preliminaries, Page 6 Section l. Real Hypersurfaces of a Complex Hyperbolic Space, Page 39 Section 2. Contact Hypersurfaces: Algebraic Consequences of the Contact Condition, Page 46 Section 3. Tubes in Complex Hyperbolic Space: the Model Hypersurf aces, Page 65 Section 4. The Lorentzian Circle Bundle over CH"(-4) and its S‘-invariant Hypersurfaces, Page 88 Section 5. Congruence and Classifications, Page l09 Section 6. An Analytic Construction of a Horosphere, Page l26 List of References, Page 140 IV INTRODUCTION The study of hypersurfaces of a given manifold is fundamental to understanding the geometric structure of its submanifolds and ultimately the intrinsic geometry of the ambient space. This has been an especially productive endeavor for hypersurfaces of spaces of constant sectional curvature and more recently for real hypersurfaces of complex manifolds that have constant holomorphic sectional curvature. In this study, real hypersurfaces of a complex hyperbolic space, Le. a complex Riemannian manifold of negative constant holomorphic sectional curvature, that satisfy certain tensor equations are classified by utilizing a Lorentzian hyperbolic SI—fiber bundle over the ambient complex space. All the hypersurfaces classified are isoparametric (have constant principal curvatures), although this hypothesis is used primarily for congruence. As a byproduct of the classification, some information is gained concerning SI-invariant hypersurfaces of Lorentzian manifolds of negative constant sectional curvature. The first condition studied is that of a real hypersurface of a complex hyperbolic space being contact with respect to the induced metric. Okumura, I19], studied this condition in l966 for real hypersurfaces of complex spaces of constant holomorphic sectional 2 curvature. Kon, [13], found a classification of contact hypersurfaces of complex projective space in terms of Takagi‘s work on isoparametric hypersurfaces of complex projective space, [27]. However, the classifications of contact real hypersurfaces as tubes occurred in [19] and then in [22), published in 1983. In these papers, a contact hypersurface of complex euclidean space is shown to be either a hypersphere or a certain type of cylinder. With the publication of [2] in 1982, Kon’s classification of contact hypersurfaces of a complex projective space can be made in terms of tubes. However, a complete classification of contact hypersurfaces of complex projective space can also be made by using the techniques of section 3 and the congruences of a sphere. For a contact hypersurface of a complex hyperbolic space the classification is given by the following theorem: MEDIA. A complete, connected contact hypersurface of a complex hyperbolic space of complex dimension n and holomorphic sectional curvature -4 is congruent to one of the following: i) a tube of radius r>0 about a totally geodesic real hyperbolic subspace of dimension n and sectional curvature -l, ii) a tube of radius r>0 about a totally geodesic complex hyperbolic subspace of complex dimension n-1 and holomorphic sectional curvature -4, iii) a geodesic hypersphere of radius r>0, or iv) a horosphere. 3 A condition related to that of a real hypersurface of a complex Riemannian manifold of constant holomorphic sectional curvature being contact is that of the induced almost contact structure A commuting with the second fundamental form H. Kon obtained a classification of real hypersurfaces satisfying this condition as well in [l3], again in terms of [2?]. Romero and Montiel, [16), found a complete classification of real hypersurfaces of a complex hyperbolic space satisfying AHzHA in l980, in terms of explicitly defined models in the Lorentzian S‘-fiber bundle over the ambient space. In our study, the classification is essentially new and is in terms of hypersurfaces of complex hyperbolic space instead of submersions of S'-fiber bundles as occurs in [16]. To wit: mm A complete connected hypersurface of a complex hyperbolic space of complex dimension n and holomorphic sectional curvature -4 whose second fundamental form commutes with the induced almost contact structure is congruent to one of the following: i) a tube of radius r>0 about a totally geodesic complex hyperbolic subspace of complex dimension p, Ospsn-I, and holomorphic sectional curvature -4, or ii) a horosphere. Semi-symmetric spaces are those whose curvature tensor annihilates itself when acting as a derivation. Nomizu, [17], in 196? and Tanno, [15), in 1969 investigated semi-symmetric hypersurfaces v‘ t 4 of euclidean space. Tanno and Takahashi, [28], widened the investigation to semi-symmetric hypersurfaces of spheres in 1970. In l969, I23], and in 197l, [24], Ryan broadened the class of ambient spaces in this type of investigation to those of constant sectional curvature. However, as far as the author knows, no work has been done on semi-symmetric hypersurfaces of an indefinite space which makes the following corollary of more than passing interest. MD”. A semi-wtric sLinvariant Waco of anti-dc Sitter spaeetimeisconguent toanSI-fiberbmdleoveroneofthe Ween ofTheoremS. Other results that are of intrinsic value themselves are generated enroute to the above theorems and corollary. For instance, the construction of the model spaces in section 3 is of interest as the technique is quite general and should yield satisfying characterizations of isoparametric hipersurfaces in other ambient spaces. In section 4, not only is the Lorentzian Sl-fiber bundle over complex hyperbolic space used to obtain information concerning hypersurfaces of the Riemannian complex space, but the Riemannian structure of complex hyperbolic space is used to obtain information concerning hmersurfaces of the Lorentzian SI-fiber bundle. In section 5, a congruence theorem for a certain type of isoparametric hypersurface in acomplex hyperbolic space is proven that enables the classification. Finally, in section 6 a horosphere in complex i VIM". 9i} 72m. II) :‘II 7-. u 1’ lo 1w.» 5 Iciai'ip Bit. .3; .r isvo slcnud wall ’8 na cl unfit; 10.3 .inI‘iUPI'iijUf‘l Inuisvm-Iif :iiiisrnmpe-im'n- A Irismpnm at am! same .3 maim'i? if) eSJb‘IMdI'JQtjl‘ .;‘ “‘ I») 3".) lie-"i397 i..‘.‘;v 3:371:99! iui' , 1'll‘f-',"".' :33 ,;' "" . i" ‘3 72V; m» iI"--i i? 4 ..,:::~ ‘0 fl :3 ‘5 0 O. i . ‘ I V‘ W‘. T J‘ , _..l‘:I.. 1 . .(b .I".) 'I Q j u. . ‘3'" ' iv . ‘. .IL" a s j? " ‘ ‘ , ' ’::,ir-;_, ’ ‘ " ‘ . ‘ i. I , ‘ ' I I I , s I. I ~ ‘ ' l 0 ’ u U E ' i \~ . -0 V .1 s ".:!.':i ' ' ' r‘r.’ ., Til (Es-m 3: 3w”; “:JLIR " +71“. " -l-. 3.. ‘° : . 7: ’.. 1 4 " ’ .“‘ .. ' ‘i ‘ ' °= ' - i . ‘ 1 ‘ , . i .1) r V I i :‘ ‘ i f' I v s _ '9 "I". .' ' . 1 u ' A. a ‘i \ J, 1: 7f ' I. ,‘i j “ft-V j r (f - t " A a . 'i '9 i ‘ I ' 5 representations of the S‘-fiber bundles over the hypersurf aces of Theorem 5. The condition of a certain direction on a real hypersurf ace of complex hyperbolic space being principal is central to the study. This condition merits an independent study, and hopefully, the following pages will facilitate such a study. 0. Geometric Preliminaries Throughout this study, all manifolds will be assumed to be smooth (C°°) and complete. C°°(M) will denote the set of smooth real valued functions ona manifold M. The base fields of all manifolds discussed here will be R (the field or real numbers) and C (the field of complex numbers). If p is a point of a manifold M, Tp(M) will designate the tangent space to M at p. which is the vector space generated by all tangent vectors to M at p. A vector field onM is a smooth assignment of a tangent vector to each point of M. Hence, on the manifolds under consideration, the set of all vector fields will at each point generate the tangent space. The set of vector fields ona manifold M will be referred to as the tangent bundle, T(M). T(M) forms a fiber bundle over M with Tp(M) as the fiber over a point pm. A subbundle of T(M) will be called a W on M. For each pelt, there is a nieghborhood U of p in M for which we can select n vector fields from T(M) with the property that the corresponding tangent vectors are linearly independent in Tq(M) at each qu. As each Tp(l‘l) is an n-dimensional real vector space. the notion of tensor applies. In the same way that a tangent vector at pen is extended to a vector field on M, we can extend the notion of a tensor on Tp(M) to that of tensor new on M. For instance. let (x, .....x nI be a subset of T(M) such that {XI ,...,X n}q is linearly independent at each point q in an open nieghborhoodU of p in it Define a positive definite inner product on each Tq(l1). w. by q=5ij (where 8 is the Kronecker delta). By extending < , >q bilinearly to the remainder of Tq(M), for each qu, we obtain a tensor field of type (0,2) on U (and consequently a local orthonormal basis of T(M)). in fact tensors of this sort are fundamental in semi-Riemamian geometry and when a certain one is selected to work with on a semi-Riemamian manifold it is often called the first fundamental form of the manifold. In many respects its choice really determines the geometry of a manifold, hence only certain bilinear tensor fields will be admissable as a first fundamental form on a given manifold. A Wbr first fundamental form) one manifold M is a non-degenerate symmetric tensor field of type (0,2)onl1, that has the property that the dimension of the negative definite subbundle of T(M) (with respect to the metric tensor) is constant onM. ‘ The dimension of the negative definite subbundle is usually ref ered to as the indexef the manifold Of course, if the index is constant then the dimensions of the positive definite and neutral subbundles will remain constant as well. If M is a semi-Riemannian manifold, then Mn'“ will denote that M is of dimension m and has index n The existence of a global metric tensor field on a manifold and its nature will determine the intrinsic geometry of the manifold. This was shown by Gauss for surfaces in R3 and by Riemam for manifolds that admit a positive definite metric tensor. The full generalization to manifolds with indefinite metrics occured under the impetus of relativity. if a manifold admits a metric tensor it is said to be W. Two important special cases will concern us: A manifold that admits a positive definite metric tensor is called a Wmanif old. Asemi-Riemannian manifold of index one is called Lgmntzian (incidentally, Lorentzian geometry is the geometry of special relativity.) The metric tensor on a manifold is used to define the lengths of vector fields and the angles between them. This allows us to speak of a local orthonormal basis of T(M), which will be called a mee, If a metric is indefinite, then there are nontrivial vectors of both negative and zerolength. We shall say that a vector field is Mike if it has positive length. tjmeLilgLif it has negative length and 11mm orneutLaL if it has length zero. The metric tensor on a manifold also can be used to measure how one vector field may vary with respect to any other as a point is moved about on the manifold. This will ultimately lead to the curvature of a manifold, so we shall define the vector rate of change 10 in any direction on a manifold: [22! initign Aggmegjjgn ona manifold M is a function V:T(l‘1)xT(M)-'T(M) that satisfies the following properties: (OJ) VIX‘QY Z=IVxZ*QVY2 (0.2) Vx(aY+bZ)=aV xY+bV x2 for all X,Y,ZeT(M), a,bcR and f,g£C°°(M). ll VXY is called the Wm Y with respect to X for the connectionV. in general there may be many different connections ona semi-Riemannian manifold M. However, we shall be interested in only one, namely the so-called Levi-Civita metric correction Ona semi-Riemannian manifold M there exists a unique connectionV such that (0.4) [X,Y]£X oY-Y ox=vxv-vyx for all X,YeT(M), ie. the connection has zero torsion, and (0.5) x=+ for all X.Y.ZeT(M). where <, > is the metric tensor field on M. lf (OD-(0.5) hold for a connectionV, then V is called the Leyl-Qiyita metric connectionon M. 12 The notion of covariant derivative of arbitrary tensor fields is crucial in defining aspects of intrinsic geometry of manifolds and the extrinsic geometry of submanifolds, as we will see. it is defined inductively as follows: Let K be a tensor field of degree (r.s) ona semi-Riemamian s manifold 11; ie. K is a multilinear mapping of TI Tx(M) into the i=l space of contravariant tensors of degree r at x for each xeM. Define the covariant derivative of K with respect to XeT(M) by s (th<)(>=Xf for all XeT(M). Vf is called the magma f in M, and is nothing more than the first covariant derivative of f with respect to the metric on M. By taking the second covariant derivative off we can obtain a (0,2) tensor on M: define Hess(f;M):T(M)xT(M)-i R by Hess(f;M)(X,Y)=---xv(t)-v xv(r)=(v 2f)(X;Y). for all X,YeT(M). Hess(f;M) is called the nessian of f onM. The existence of tensors that are covariant constant will force certain geometric and topological consequencesona manifold. as we will see later in this survey section. An easy example of a tensor that is covariant constant is that of a constant function on a manifold M. Clearly, if f is constant onM then for any vector field V we must have 0=Vf=VVf. (This is easily verified using the partial differential operators obtained f om a local coordinate system as a l4 local orthonormal basis of T(M).) Conversely, if va =Vf =0 for all VeT(M) it is not hard to show that f is constant on M. in general we say that a tensor K onM is paLaLlLl if VK=0. Notice that (0.5) says that the Levi-Civita metric tensor of a semi-Riemannian manifold is parallel. The idea of "straight" in a semi-Riemannian manifold is closely ‘ Iirked to the notion of covariant constant. For instance, given any line L in R" and ch, there is a mit tangent vector V in the direction of one of the rays emanating from p along L. Translating the origin of Rh to p, we can choose a coordinate system of Rh in such a way that L is a coordinate axis, say spantxtl. We may then set V=8l8x1 so that vvv=o. The notion of “straight” is also related to the idea of distance in a semi-Riemannian manifold. in an arbitrary manifold, a length minimizing curve is called geodesic Let o:[0,rI-iM be a curve in a semi-Riemannian manifold M that is parameterized by arclength. Let V=O’(t) be its velocity vector field at 0(t) for any t6[0,r]. if a is geodesic, then its acceleration must vanish, i.e. o”(t)=0. Using this IS and the trivial observation of the previous paragraph as motivation we shall say that a curve a is geodesic if VVV=0 along 0 where V is the velocity vector field of 0; Le. a curve 0 is geodesic if its velocity vector field is covariant constant with respect to itself. For each peM". there exist n geodesics through p that are mutually orthogonal at p. Then the velocity vectors of these geodesics at p are mutually orthogonal tangent vectors and hence form a basis for TD(M). Conversely, given an orthonormal basis of Tp(M), there exist it corresponding geodesics whose velocity vectors at p are the elements of the given basis. As the geodesics are the means of measuring distances in a semi-Riemamian manifold we see that a semi-Riemannian manifold is approximated by its tangent spaces. That is, given any point peM, there exists a neighborhood of the origin in Tp(M) that is diff eomorphic to a neighborhood of p in M. Not surprisingly. a dif f eomorphism can be defined in terms of geodesics. 16 Q2! injtign Let peM and X be a unit vector in Tp(M). Let 25(t) be the geodesic emanating from p with velocity vector X at p (i.e. b”(0)=X and b(0)=p), with domain (a,b). Set expp(rX)=b'(r) for r£(a,b). expp carries lines through the origin in Tp(M) to geodesics through peM. Thus. distances in M near p are approximated by distances in Tp(M). expp is a convenient tool for discussing semi-Riemannian analogues of spheres and tubes (as is done in section 3). Although Tp(M) approximates a neighborhood of peM, the approximation is in general not very good; that is, the neighborhood may have to be of very small diameter in order to acheive a given degree of accuracy. This is a manifestation of the intrinsic geometry M. in particular, the degree of accuracy will depend upon the curvature of M at p. The curvature of a curve in R2 or a surface in R3 is easily understood intuitively. However, generalizing this concept to l7 Riemannian and semi-Riemamian manifolds requires the definition of a new tensor field: E f. 'I'0 Let M be a semi-Riemannian manifold with Levi-Civita connection V. The W onM is defined to be a tensor field B of type (1,3) given by: R(X,Y)Z=V XVY Z-VY VxZ-ley] Z for all X,Y,ZeT(M). R seems far removed from the usual idea of Gaussian curvature of curves and surf aces, but is a necessary generalization to discuss the curvature in semi-Riemannian manifolds of arbitrary dimension However, this generalization gives rise to more than just one notion of curvature. The notion that generalizes Gaussian curvature is that of sectional curvature. 18 Qefinjtigg Let X.Y be linearly independent elements of T(M) where M is semi-Riemannian manifold with metric tensor <, >. The mm W of the plane Il=span{X,Y} onM is given by K(Il)= /t-21. in case X and Y are orthonormal, K(II)=. Not surprisingly, K(TI) is a geometric invariant, i.e. K(TI) is independent of the choice of basis of II. It is not hard to verify that for a surface in R3, the Gaussian curvature agrees with the sectional curvature. lf K(Ilp)=K is constant for any choice of non-degenerate plane P section lip at p, then M is said to have constant sectional curvature at p. F. Schur showed that if M is a corrected semi-Riemannian manifold that has constant sectional curvature at each point p of M, then Kp is a constant over M; i.e. Kp=K for all peM. In this case, M is said to be of constant sectional curvature K. lllib 19 For example, a euclidean space with a semi-Riemannian metric tensor and Levi-Civita metric connectionhas K(ll)=0 for every plane distribution 11. As such, an is said to be flat, that is, a space of constant sectional curvature 0. If we consider the sphere S“(r'2)={(x 0,...,x n)lX02*"12*---”‘ n2=r2). with the metric induced by the ambient euclidean space, it is not hard to show that K(TI)=r'2 for every plane distribution II on 5")(r'2). This is an example of a space of positive constant sectional curvature r’z. The hyperboloid n HW-r’2)={(x0,x' ..... X n)|'X 02*2 xi2=-r2} l=l endowed with the metric n n d52=IZ dxideil/ti- ('l4r2)2 (x9e i=0 i=0 is an example of a space of constant negative sectional curvature -r-z 20 Complete, simply connected spaces of constant sectional curvature are called W In [32]. any Riemannian space form is shown to be isometric to one of the examples in the previous paragraph as part of a classification of semi-Riemannian space forms. We shall have the opportmity to work with the semi-Riemannian euclidean space R220“) equipped with the metric 2M “'9’” 090'“i 91 ‘2 “tilt i=2 and an imbedded Lorentzian htpersurf ace of 822"“) szm' ('I)={(X 0,8, ,...,X 20’2” ’802‘3' 2+8 22*“!!! 2m22="l}. (with the appropriate metric, RFD" (-l) is a Lorentzian space form of sectional curvature -l). A real space form will have a particularly simple form for its curvature tensor. lf M(c) has constant sectional curvature c, then R(X,Y)Z=C(X-Y). 2| (As we already knew, the curvature tensor of a euclidean space vanishes.) On S“(l), R(X,Y)Z=X-Y and on H“(-l) R(X,Y)Z=Y-X. If we compute the covariant derivative of R ona space form we will find that VR=0. So constant sectional curvature is linked to the idea of covariant constancy. In general we will say that a semi-Riemannian space form is locallysymmemmf its curvature tensor is parallel. Thus. local symmetry is a generalization of constant sectional curvature. Local symmetry has a generalization, as well. Define a new tensor field R-R by letting R act on itself as a derivation let X,YeT(M). Def ine a (1,3) tensor field R(X,Y)-R by setting 22 (R(X,Y)-R)(V,W)Z=IR(X,Y),R(V,W)IZ-R(R(X,Y)V,W)Z-R(V,R(X,Y)W)Z for all V,W,ZeT(M). Applying (0.4) and the definition of R, we find that VR=0 implies that R(X,Y)-R=0 for all X,YeT(M), or simply RoR=0. A manifold whose curvature tensor satisfies R-R=0 is said to be W ([25]). Hence, semi-symmetry is a generalization of local symmetry. A great deal of research has been performed on semi-symmetric Riemannian manifolds, (see [15], [17], [231-[26] and [ZBI-BOI), but little if any on semi-symmetric Lorentzian spaces as is done in section 4. All of the preceeding geometric concepts are aspects of what is called the intrinsic geometry of a semi-Riemannian manifold, as they arise from the structures intrinsic to the manifold and not from any external considerations. However. by immersing one semi—Riemannian manifold into another, we can study the geometry of the image of the immersion as viewed from the ambient manifold. This geometry is called the extrinsic geometry of the immersion An immersed manifold in a semi-Riemamian manifold is called a 23 submanifold. (We shall blur the distinction between an immersion and its image.) It must forma semi-Riemannian manifold with structures compatible with those on the ambient manifold; e.g. the metric induced by the ambient manifold forms a semi-Riemannian metric on the submanif old with index at most that of the ambient manifold. As the extrinsic geometry of a submanifold is intricately linked I to the intrinsic geometries of the ambient space and the submanifold, it can be determined by “comparing" the two, i.e. by finding mathematical relations between similar aspects of the two geometries. Hence, given information about either the intrinsic or extrinsic geometry of a submanif old or the intrinsic geometry of the ambient manifold, one can usually determine some information concerning an unknown geometry. Let M be an immersed semi—Riemannian smmanif old of a semi-Riemannian manifold M. The Levi-Civita connection? on F1 will induce a connectionv onM that is Levi-Civita with respect to the metric induced onM by F1. For X,YeT(M) we can write 24 (0.6) vxv=vxv+s(x,v) where the first term is the component tangent to M and the second term is normal. 8 forms a symmetric. normal-valued. bilinear form on T(M)xT(M) and is called the W of the submanif old M. (0.6) is referred to as the W for M in F1. _ If I: is a normal field to M in T(M). then we can write the Wmforflin Ii: (0.7) Vx£=-A£X+V1x£ where as before the first term is tangential and the second normal to M. A: is called the W associated to I; and forms a self -adjoint tangent bundle endomorphism on T(M). V1 is called the W onM and actually satisfies all the axioms of a connection if tangent vectors and normal vectors are used appropriately ([3]). Due to the unique nature of B, its covariant 2S derivative. {78. has a separate definition and is defined (as in [3]) by (vaXVJ) =V‘X(B(V.Z))-B( vxvz)-B(v.v x2) for all X,Y.ZeT(M). 6 is called the comeclionof Van der Waerden-Bortolotti. The tensors B:T(M)xT(M)-9T(M)1 and A:T(M)xT(M)*-iT(M) obtained by (0.6) and (0.7) contain all the information necessary to determine the extrinsic geometry of M in F1, and are intimately related via (0.8) <£,B(X,Y)>==< RIX,Y)Z.W> +- 26 and the Cedazzieeuatjen (0.10) (R(X,V)Z) 1.-.(ttxa)(t,z)-(t't,,a)(x,z) for all X,Y,Z,WeT(M), where 1 denotes the normal component relative to M. in these equations we see that the second fundamental form provides a measure of the difference between the curvature of the submanif old and that of the ambient space. In case the ambient manifold is a space form M(k), the Gauss and Codazzi equations are simpler and more explicit: = k(-) +- and (vYB)(x,2)=(exB)(v,2) 27 for all X,Y,Z,WeT(M). The second fundamental form can be used to def lne a normal field that is, in a sense, a measure of how M curves relative to M. Let X] ,...,X n be a local orthonormal basis of T(M) consisting of non-neutral vector fields and set (i=1 if Xi is spacelike and ei=-l if Xi is timelike. The normal field E"=(I/n)z €iB(Xi,Xi) is called the Wot M in F1. It will lengthen and twist in T(M) according to the relative curvature of M. M is said to be mjgjmaL in M if E" vanishes onM. Straightness and distance are concepts that serve as useful tools for comparing the intrinsic and extrinsic geometry of a submanif old. For example, geodesics in 52(r), r>0, are great circles isometric to S'(r) which are certainly not geodesic in the ambient space R3. Hence. 32(r) is not I lat in R3 althougi locally it may 28 .. appear to be so. in contrast, an extrinsically f lat surface in R3 is forced to be a plane as all its geodesics would have to be extrinsically straight, that is, lines. The notion of extrinsic flatness is formulated as follows: E ["1' If all the geodesics of a submanif old of a semi-Riemannian manifold are also geodesic in the ambient manifold, then the submanif old is said to be mm It is easy to see geometrically that the totally geodesic submanifolds of a euclidean space are euclidean spaces of lower dimensions, and that the totally geodesic submanifolds of spheres are merely great spheres of lower dimensions. However, for spaces in which geometric intuition f ails us, it would be nice to have an analytic criterion of this condition vlt happens that there is a nice characterization of this in terms of the second fundamental form. in a semi-Riemamian manifold. it can be shown that a submanifold is 29 totally geodesic if and only if its second fundamental form vanishes. From these examples we catcha glimpse of why the intrinsic geometry of a manifold is determined by its totally geodesic submanifolds. On the other side of the coin, we can see that the length of the second fundamental form serves as a measure of how far a submanifold deviates from being extrinsically flat and the length of the mean curvature vector serves as a measure of how far the submanifold deviates from being minimal. From (0.8) we see that for a vector field E normal to a totally geodesic submanif old M. the associated Weingarten map vanishes. i.e. Afo‘le) This is an example of a more general condition that can be imposed ona submanifold, that of requiring the existence of a normal field to have an associated Weingarten map that is proportional to the identity map on T(M). in general, a normal field is said to be ymblei; onM if A: is proportional to the identity map on T(M). lf B(X,Y)=£ M for all X,YeT(M), then M is said to be totally umpiilig in the ambient semi-Riemannian space. 30 For example, a sphere 32(r) of radius 1/ I r in R3 is totally umbillic with a single global Weingarten map A£=rlT(52(r» (Notice that there is only one choice f art, on 52(r) and that {452(0) in general, S"(r) is totally umbillic in R”, m>n Small spheres Sm(r’) are also totally umbillic in S"(r), for m. Hence, almost all the information concerning the extrinsic geometry of the hypersurf ace is embodied in the single (1.1) tensor H. Henceforth. we shall refer to H as the second fundamental form of the hypersurf ace. The Gauss and Codazzi equations for a hypersurf ace M of a real space form M(k) are: R(X,Y)Z=k(X-Y)+HX-HY and for all X,Y,ZeT(M), which means that the curvature of a hypersurf ace is (relatively) easy to analyse. We immediately see that the task of determining the extrinsic geometry of a hypersurf ace reduces to 32 analysing the behavior of the T(M) endomorphism H. or course, many different types of hypersurf aces of Riemamian space forms have already been classified. However, this is not at all the case where the ambient space form is semi-Riemamian or in particular if the ambient space form is merely Lorentzian The primary reason for this is that H, although self -adjoint, may not be diagonaiizable (i.e. have real eigenvalues) on T(M) if M has non-zero index; a stark contrast to the Riemamian case. In sections 4 and 5, an initial attack on this problem is made when semi-symmetric Lorentzian hypersurf aces of a certain ambient Lorentzian space form are classified. However, the main purpose of the subsequent sections is to classify some hypersurf aces in a certain Riemamian manifold that does not have constant sectional curvature. Yet, this ambient space will have a specific notion of curvature being constant if we view it as a manifold with base field C instead of B. So a short discussion of complex manifolds should ensue. Let M be a 2n-dimensionai Riemamian manifold with JeEnd(T(M)) .__.a 33 “ that has the property that J2="T(M)° Then at each point peM, TD(M) forms a complex vector space that is isomorphic to Cn and J can be associated with the endomorphism obtained on CPERZ", viewed as a real vector space, from multiplication by i=l-i. J is called an almost complex structure on M. Clearly, if sucha structure exists on a Riemannian manifold, the manifold is necessarily even-dimensional. Of course. Cr)“ itself forms such a manifold. in fact. C'M can lead us to the correct generalization of semi-Riemamian manifolds overR to those with base field C: by defining a symmetric bilinear form . q n F q(z,w)=-Z sz j +2 zkv'vk i=0 k=q+l for z=(zo.z, .....z n) and w=(w 0m, .....w n). we obtaln a non-degenerate Hermitian metric on CM that turns Cn+1 into a semi-Riemannian complex manifold of index q. C““1 with the metric Fq will usually be written an”. Cqm‘ can be made into a real even-dimensional 34 semi-Riemannian euclidean space qum2 by using as a metric the (0,2) tensor < , >=Re(Fq). Now the almost complex structure J is Hermitian with respect to < , >, that is = for all real vector fields X and Y on qumz, so J is both orthogonal and skew-adjoint. Furthermore, J will be parallel with respect to the Levi-Civita connection induced by this metric. qumz is an example of what is called a Kaehler semi-Riemannian manifold. E [. T A semi-Riemamian manifold M with an almost complex structure J and metric <, > is Kaehler provided that i) = for all X,Y£M, and ii) VJ=O. 35 On a Kaehler manifold, J is often referred to simply as a complex SII'UCIUI'Q. Other examples can DB derived from anH . For instance, consider the sphere 52"+1 (1)={ze CO” | Izol2+|z, [2+...+|z n|2=l}. We can form a Riemannian submersion of 52”“ (1) onto an complex n-dimensional Riemamian manifold CPn by identifying all points on 52’“1 (1) that lie ona complex line through the origin of CM . The metric and almost complex structure of CPn can be induced from the natural complex structure of Cr)“. Notice that 52ml (l) forms a principal fiber bundle over CP" with fiber S'(l). CP" is called complex projective space. If M is a Kaehler manifold, then for any pcM and XeT(M), the plane Tip=spanIXp,JX D} is invariant under J and is said to be a holomorphic section at p. The sectional curvature K(Tlp) is called the holomorphic 36 sectional curvature of M by II at p. If the sectional curvature is a constant for all J-invariant planes Tlp at peM, i.e. K(llp)=Kp for all holomorphic sections lip, then M is said to be of constant holomorphic sectional curvature at p. As in the real case, it is well known that if M is of constant holomorphic sectional curvature Kp at each peM, then Kp is a constant over M. in this case M is said to be of constant holomorphic sectional curvature. if, in addition, M is simply connected, M is called a complex space form. Cqn+1 is a complex space form of zero holomorphic sectional curvature whereas onCPn a metric (namely the Fubini-Study metric tensor) can be constructed that turns CPn into a complex space form of holomorphic sectional curvature 4. CPn is compact and has diameter n/Z under this metric. The primary ambient space in the following sections is a complex hyperbolic space, CH“, that has constant holomorphic sectional curvature -4. Sections 1, 2 and 3 only use the abstract properties derived from the constant holomorphic sectional curvature of CH". However, CHn can also be 37 constructed in a fashion similar to that of CF", only in this case the submersion is from a Lorentzian hyperbolic space f orm. This construction is crucial to obtaining the geometric results of the later sections. As with real space forms, complex space forms and their submanifolds admit relatively simple expressions for their curvature tensors. if M(c) is a complex space form of holomorphic sectional curvature c, then the curvature tensor R of M is given by (O.l2) R(X,Y)Z=(C/4)[X-Y +Jx-JY+2JZl for all X,Y,Z£T(M). if M is a real semi-Riemamian manifold immersed in a complex space form M(c) of complex dimension n then the Gauss and Weingarten formulae hold for M as a submanifold of the real 2n-dimensional semi-Riemamian manifold M. Now (0.9) and (0.10) can be combined with (0.12) to obtain the Gauss and Codazzi 36 equations of M in M. In the sequel we shall be interested in the particular case where M is a real hypersurf ace of the Riemannian complex hyperbolic space form CHn(-4). 39 1. Real Hypersurf aces of CH'Y-zi) Let CH“(-4), n22, denote a complex hyperbolic space with the Bergman metric tensor, i.e. a complex space form of constant holomorphic sectional curvature -4. Let "Zn-l be a real hypersurf ace of CH“, V and V be the metric connections onM and CH", respectively, so that the Gauss and Weingarten formulae can be written as: (1.1) va = vxv+i; , w; = -HX for all X,Y£T(M), where I’, is a unit normal field on M in CHn and H denotes the second fundamental form (in this case the Weingarten map of I; in End[T(M)l). We shall refer to the eigenvalues and eigenvectors of H in R and T(M), respectively, as principal curvatures and principal directions. if J is the complex structure of the ambient complex space form, it induces an endomorphism cl» of rank 2n-2 and a linear functional f on T(M) given by setting at each point p of M (1.2) JX = tX+r(x)a for all X in Tp(M). Set U = -J£. AsM is of codimension one we have U£T(M). The following equations now hold for all X,Y in T(M): (1.3) f(X) = (1.4) mix) = 0 (1.5) 410 = 0 (1.6) izx = -X+r(x)u (1.7) = - (1.8) <4sx, = —t(x)r(v). (¢,f,U) is an example of what is called an almost contact structure 41 on M. The tensor fields ‘i’ and U have the f oliowing derivatives: (1.9) vxu = tux (1.10) (inW = f(Y)HX-U. We also have the usual Gauss and Codazzi equations for a real hypersurf ace of a complex space form (of holomorphic sectional curvature -4) in terms of 4) and H: (1.11) R(X,Y)Z = v-x+¢v-ix-2iz +HX-HY (1.12) (VXH)Y-(VYH)X = -1(X)¢v+r(v)¢x-2u for all X,Y,ZcT(M), where R is the curvature tensor onM. An important special case for us will be when U is a principal 42 direction on M. Under this assumption, more information can be gained concerningthe structure of M. For example, by applying (1.9) and assuming that HU=o = = (xtx)r(Y)+ot-. Now using (1.4) and (1.12). <(VxH)Y-(VYH)X,U> = -2v>. Combining these equations yields -2=(x0t)t(v)+o<- -(Yo<)r(x)-tx+ =(x::)t(Y)-(v«)r(X)+o<<(iH+H¢)x,v>-2 where (1.7) is used to combine terms. Rewrite the above equation as (1.14) 2<(Ht11+ti)x,v>=(xo:)t(v)-(vo:)r(X)+«<(¢H+H¢)x,v>. Replacing X by U and then Y by U in (1.14) yields Yo<=(Uo<)f (Y) and Xo<=(Uo<)f(X). Substituting these values back into (1.14) gives 45 2<(H¢H+4>)x.v>=(Uo<)1(x)1(Y)-(Uot)t(Y)t(x)+o<<(¢H+H4»)x,v> =o<<(¢H+H)x,v>. As X and Y are arbitrary we have the assertion // Once U is principal it is clear that we can extend to a local orthonormal basis of principal directions ix, ..... X 2n-2:U} of T(M) with (X, ,...,X 20-2} forming a local orthonormal basis of ker(f). Assume that each Xi has as principal curvature hi, i=1,...,2n-2. The next question to ask, since ker(f) is tit-invariant, are there nontrivial i-invariant subspaces of ker(f) that are also H-invariant? Not surprisingly, the answer is yes. Lemma-[I41 If A is a principal curvature on M, let 0). denote the distribution of principal directions onM with principal curvature A. if XernkerU) and 2.2-110, then AX is also principal. Proof: 2(Hi>H+)x=2>.H><+2t»x and a(¢H+Hi)x=ax«iX+aH¢x So by lemma 1, H4>x=(ot then 2k2-2=0 as X and hence 0X can be chosen to be nontrivial in ker(f). This is equivalent to 99:1 and «2=4. // For the remaining case assume first that 062 and 25-1. Then, for X504, (1.13) yields 0X principal with principal curvature 1. However, if 062 and 951. then for X60], (1.13) is an identity on the distribution span{X,4>X,U}. A major consequenceof Lemma 2 is that whenever U is principal on M, we can choose a frame [X] ,...,X ”-1 ,ix, .....tx n-i .U) on M that 47 consists entirely of principal directions with the property that A interchanges the distributions spmiX' ,...,X ".1 } and spahiixI ,...,ix n.1}. The set {xI ,...x ".1 ,ix, ,...,ix ".1 } forms a local basis of the distribution ker(f) onwhich A acts as a complex structure. The principal curvatures of a p-invariant plane span{Xi,¢Xi} will be related by (1.15) Ui=(00\1"2)/(2Ai‘0() where HXi=xiXi and Hixi=31ixio whenever «12%,. The principal curvatures need not be constant even if U is principal as we will see in example 4 of section 3. This is contrary to the situation of the ambient space being CPn (see [14]). However. there are two classes of real hypersurf aces in CHn that do have U principal with all the principal curvatures constant and these are the subjects of section 2. 48 2. Contact Hypersurf aces: Algebraic Consequences of the Contact Condition Let M be a 2n-l dimensional Riemannian manifold that admits a triple of tensor fields (try), (where ieEnd[T(M)l, f is a linear functional on T(M) and UeT(M)), satisfying (1.3) and (1.6). As remarked in the preceeding section, such a triple (¢,f,U) forms an almost contact structure on M. in general, a Riemamian manifold that admits an almost contact structure also admits a metric satisfying (1.8). From these formulae (1.4). (1.5) and (1.7) canbe obtained. This is a generalization of another intrinsic condition that can be imposed on a Riemamian manifold: M is said to be a contact manifoldif it admits a linear functional f that satisfies f «(df)”' ‘ =0. (Such a manifold also admits an almost contact structure (tau), Ill.) In section 1, we saw that a real hypersurf ace M of CHn (in fact of any complex space form) automatically admits an almost contact structure that is already compatible with the metric induced from Qr' 49 ~ the ambient space. in [19], Okumura showed that if MZ'H is a contact real hypersurf ace of a complex space form of complex dimension n, then (2.1) ¢H+H<) to see that 5| 0=H2X-2pHX+(o, where Vo: denotes the gradient field of the function o< on M, we have Votcspan{U}. Thus, Vo<=U=(Uo<)U. Hence, 52 VX(V0<)=X(U0<)U+(U0<)\;=x(uc<)t(Y)+(utx) and similarly =v(uo<)r(x)+(uo:)<¢1iv,x>. But, we also know that (VX(VO(),Y>:X’=Y(Xo<)-V Y X(o<). Then, -=X(Yo<)'Y(Xo<)+V Y X(o<)-VXY(0<) =IX.YI(0<)+IY.XI(0<) =0. Now 0=‘ =x(uo<)1(v)+(uo<)<¢Hx,v>-v(Uot)t(x)-(uot)t1v,x> which yields (2.3) x(uo:)t(Y)-v(uo:)r(X)=(uo:)-(uot)< =(uo<)<(i11+11i)v,x> 54 =2p(utx). Substituting UforY and then forX in (2.3) gives X(Uo<)=U(Uo<)f (X) and Y(Uo<)=U(Uo<)f (Y). Substituting these values into the left hand side of (2.3) yields o=2p(utx). Choosing X=¢Y=0 shows that Uo<=O.// Thus, M has at most three distinct principal curvatures, all of which must be constant by (2.2) and Lemma 4. Since CH“(-4) has no 55 complete totally umbillic hypersurf aces (see [5]). we are left with only two cases to consider: A)(2.2) has two distinct solutions x12)? or B) (2.2) has only one solution Mot. Case B) is the easiest to analyse. Let D). and Do, denote the eigendistributions of it and o< on M. or course. 0,‘ is of dimension 2n—2 and 0‘x has dimension 1. It is immediate that A acts as a complex structure on D)‘. Requiring (2.2) to have only one solution forces 2t=p and pz-otp+l=0. The latter equation has real solutions only when 03-420. in case o<2-4>0. we may regard at as a parameter. By selecting the orientation of M appropriately, we may assume that o<=2coth(2r) and A=p=tanh(r) or coth(r), for some r>0. Otherwise, set o<=2 and A=p=l. So with respect to the frame consisting of principal directions {Xi ,...,X n-i ,AX, ,...,AX n-l ,U} of T(M), (where for 56 each i=1,...,n-l, HXi=XXI and HAXi=xAXi), H has only three possible matrix representations: i) diag(tanh(r)l 2n-2,2coth(2r)), ii) diag(coth(r)l2W2,2coth(2r)), or III) diag(l 20-22) Notice that in each of these cases, HX=AX+o=x- =- :-xl (Y,¢X> =0. This shows that VxYeker(f) so that HVXYeker(f). Let ZeKer(f). Then :'<(v xH)Y,Z> 3}] ' =x, -U> =x, - 59 3A1 (VXY,Z>"+"A I +AI . Thus, HVXY=A1VXY which completes the proof of step 1. Step 2- If XeDl and YcD 2, then VxYeD zospanIU} and VYXEDI OspanIU}. Proof of step 2: Let ZeDI . Then, =0 => o=+ ... =o by step 1. This yields the first inclusion The second follows in exactly the same way by choosing 2602. 60 Step 3- If X60, and Y60 ZnIAXP, then VXY602 and VY XeDl. (Here we see the reason for the stipulation n23.) Proof of step 3: From the hypotheses. =X- =- :-)‘I (Y.¢X> =0. Combining this with step 2 we have step 3. Let xw, and 11:0 ZnIAXP with |va|=1 Applying steps 1,2 and 3 we find that R(X,Y)Y=V X(VY Y)‘V Y (VxY)"V [X,Y] YED 2.5DaI'IIU} 6| by writing [X,Y]=V xY-VYX. However, a direct computation using the Gauss equation reveals that R(X,Y)Y=(>t '2‘2-1)X60 1- Therefore, R(X,Y)Y=0 and since X is nontrivial we must have A, x2-1=0. Now if o<=>ti for i=1 or 2. the same statements hold if D, and 02 are replaced by Di nker(f) and Dznker(f). // Because of (2.2) we must have it] 22=o0, i.e. when 0(2’4<0. Notice that o<=O is ruled out by o0. Then the solutions of (2.2) are A] =tanh(r) and x2=coth(r). (Note that p=(tarlh(r)+coth(r))/ 2 in this case.) So with respect to a suitably chosen basis of T(M)=D, oozospanIU}, H has the matrix representation 62 diag(tanh(r)l “.1 ,coth(r)ln_l ,2tanh(2r)). Remerg: M. Okumura in [19] treats the case «=11- f or i=l or 2 as a separate case. But from our work so far we see that this occursin case A) for a specific r, namely r=ln(2+f 3) so that o<=it2=l3 and Aquick glance at the classification results in [16] will convince the reader that not all hypersurfaces satisfying (2.4) AH=HA are contact. Yet, in section 5 we will still be able to obtain the same sort of characterization for these hypersurf aces as In the contact case. Also in section 4 we shall analyse hypersurf aces of a Lorentzian space that submerse into hypersurf aces of CHn satisfying (2.4). So we should spend a little time on algebraic consequences of 63 (2.4). (For more detail see [16].) Combining (2.4) with (1.4) yields Uprincipal: say with principal curvature ot. An argument similar to that of Lemma 4 shows that o: is constant. By combining Lemma 1 with (2.4) we have (H I ker(t))2'°0 we can again regard o< as a parameter and set «=2coth(2r), r>0. The solutions of (2.5) are now x=tanh(r) or coth(r). In case «=22 set )t=:1. if D)‘ is a proper subspace of ker(f), then (2.4) ensures that D,‘ is 64 A-invariant. Since ker(f)=D)‘ODI/,‘, Di/Jt is also A-invariant so that A acts as a complex structure on each of the even dimensional distributions D,‘ and 01/): . From our analysis of case A) for contact hypersurf aces we see that a hypersurf ace satisfying (2.4) is not in general contact. The possible matrix representations for the second fundamental form on real hypersurfaces satisfying (2.4) with respect to a suitably chosen basis of T(M)=D)‘ODIA ospanIU} are now i) diag(tanh(r)l 2p.coth(r)l2n-2_2p.2coth(2r)). p=0 ..... n-l.or II) diag(l 20‘2’2)‘ Of course, if p=0 or p=n-l in i) then M is contact. ii) is obviously CORISCI. 65 3. Tubes in Complex Hyperbolic Space: the Model Hypersurf aces In this section hypersurf aces of CH”(-4) are constructed that have second fundamental forms with the algebraic properties set forth in section 2. Hence, we are provided with an ample supply of contact hypersurf aces as well as those satisfying (2.4). Our initial discussion will be of a more general nature: tubes in Riemamian manifolds. (For more detail, see [6], [9], [11] and [311) Recall first the notions of cut point and cut locus. (A detailed and analytic discussion of cut loci can be found in Vol 11 of [121.) A cut point of a point p in a Riemamian manifold M is a point c=b’(t), where 21' is a geodesic emanating from p=b’(0), with the property that for s>t, the length of the curve b’(J), J=I0,sl, is greater than the distance d"(p,2f(s)). For instance, if peSz(r), its only cut point is its antipodal point. The smuggle of a point peM, written Cut(p), is the set of all cut points of p. The cut locus of a point on a sphere is a singleton whereas for a point p on a cylinder, Cut(p) is the axial line opposite 66 1). Define C(p)=min{d(p.q) I quut(p)}. Let Nm be an immersed submanifOld of a Riemannian manifold M“. Define the unit normal sphere bundle of N by: S‘(N)={X£T(N)‘: |x|=1}. Set c(N)=ianc(p) | peN}. Now for each re(0,c(N)), define the IIIQLQI regime Lebeyjfljnfl to be the hypersurf ace given by Nr={equ(rx): qu, XES‘(N)}. Let c(t,X g) be parallel translation of vectorfields along the geodesic 67 (3x)qzt-¢exp q(tX). For p=equ(rx)£Nr, c(t,X q):Tq(M)-*TD(M) is a linear isometry. Because parallel translation preserves the fibers of vector bundles in a Riemamian manifold. ([20). p.66), we have (3.1) Tp(Nr)=t:(r.>0, by axe): q=c(t,X q)"I {R(c(t,X q)1 q,e(t,x q)Xq)1:(t,X q)qu where R is the curvature tensor of M As we are primarily interested 6B in the tangent space of the tube Nr, set Finally, define F(t,X)eEnd({X) 1), for each XeS‘(N) to be the solution of the initial value problem: (3.3) (dz/at 2)Irttx q)I+Fix(t)oF(t,X q)=o F (0,Xq)=P (d/thF(I,X q)] I t:0:"AXOP'.'Pl for each qu, where P:{X)‘-tT(N) and P1:IX)1-’T(N)10{X}‘l are orthogonal projections of the vector bundle {X}‘=T(N)e IT(N)‘0{X}‘I onto the indicated component distributions, and Ax is the Weingarten 69 map of X onN in M. To simplify notation we shall write F '(s,X q) for (d/dt)[F(t,X q)l I H and F”(s.X q) for (02/012)IF(t.x q)l I tzs. lbeecemI-Ilil The second fundamental form of Nr at p=equ(rx) Is given by (3.4) Hr=e(r,Xq)oF'(r,Xq)oF(r,Xq)" .e(r,xq)'l. // Hence, in order to find an explicit representation of the second fundamental form of a tube. we need merely select a suitable basis of MN) using (3.0 and (3.2). solve (3.3) and then compute (3.4). Of course (3.4) says that HreEndIT(Nr)] at p=equ(rx) is nothing more than parallel displacement of the endomorphism F ’(r,X q)oF(r,X q)“I eEndi{Xq)‘I 70 along the geodesic 3X emanating from q and passing through p. Now we will assume that M is not an arbitrary Riemamian manifold, but the ambient space discussed in sections 1 and 2, namely a complex hyperbolic space. Let N be an immersed submanifold of CH“(-4). As CHn is a symmetric space, oncea suitable basis of {qu1 is selected (where qu and XeS‘(N)), parallel displacement along the geodesic fix will preserve the basis and the respective orthogonality relations between its elements. Furthermore, Hr will have the same matrix representation with respect to the displaced basis as F’(r,X q)oF(r,X q)" has with respect to the chosen basis of {Xq}1. This simplifies the calculation of (3.4) considerably. An additional feature of CHn is that Rx(r) is of a particularly simple form. Let XeS*(N) and YeIX.JX} 1. Direct computations using the Gauss equation show that: 7i (exam q=::(1.x qu IR(c(t,X q)Y q,c(t,X q)>< q)x q1 =t:(t,X q)“I [-‘t:(t,X q)1! qI :-Yq and (Fix(t)JX) q=e(t,X q)'I [R('t:(t,X q)qu,e(t,x q)Xq)e(t,X q)qu =t(t,X qrI [-47:(t,X q)qul =-4qu for all t>0. In conjunction with the following observations the task of computing a representation of Hr will be greatly simplified. Asa hypersurf ace, Nr has a single well defined global unit normal It. From the earlier discussion on tubes, at any point p=equ(rX)eNr, we can write £p=e(r,Xq)Xq. In this way a unique point q and a unique 72 direction in S‘(Nq) can be associated to each point peNr. In order to simplify notation set Y"=t:(r,X q)YETp(Nr) for any YeIX qil. In particular, we shall write 1;" for the global normal on Nr, and I, will refer to the associated direction in S‘(N); i.e. {*p=equ(r£q). ’— f i L3,...— ” «is! of N In terms of section 1, U”p=-J€," p=-t(r,Xq)J£ q and 73 twptlntpthnrtrtpretend:qr) . (Ith.£q}*an(N)*)l is the A-invariant subspace, ker(f), of Tp(Nr). Also, CH”(-4) as a space of constant negative holomorphic sectional curvature, is a space of negative sectional curvature. Since CH”(-4) is simply connected, by Theorem 8.1, Chapter VIII, Vol.- 2 of [12]. all cut loci will be empty. This means that tubes of any radii may be constructed about smooth submanifolds. In particular, a certain class of submanifolds will give us tubes that are hyperswf aces of the type discussed in section 1. Proposition 1- Atube around a proper totally geodesic submmif old of CHn is a hypersurf ace that has U as a principal direction Proof: Let N'“ be a totally geodesic submanifold of CHn of dimension m<2n Let Nr be the tube of radius r about N and p=equ(r£q)£Nr. 74 From the previous discussion we can select a local orthonormal basis Bq of {liq}l that has Uq=-JI;q as an element and contains a basis of Tq(N). By Theorem 1 of I51, N is either a totally real or a complex submanifold of CH". Hence. Uq is either tangential or normal to N. Thus, R110), P and P1 each have diagonal matrix representations with I respect to B . Because A50, (3.3) is now q (3.5) i) F"(t.{ q)+R£(t)oF(t.£ q)=o, t£(0,rl ii) F(0,Eq)=P and iii) F’(0,£q)=P". Order Bq in such a way that R50) is represented by the matrix diag(-12n_2,-4). We shall regard (3.5) as a matrix valued differential equation and will write its matrix solution as F(t,£, Q)=[f” (t)] where 75 i,j=l,...,2n-l. 50 (3.5) i) yields f "i j (t)-f i j (t)=0 for i=2n-i, and I”2n_l’j (I)'4I 2n_1.j (0:0 OII'IBI'WISB. These ordinary differential equations have solutions of the form: fij(t)=aije't+bijet for i=2n-l, and - -2t 2t - f 2n—l,j ’32n-l,j e $20.” 2 OII'IBTWISB, where the a” ’s and the Pl] '5 are constants with respect to t. Both P and P1 are diagonal with 0's and 1's onthe diagonal (of course: P+P‘=I2n_' ). Thus, from (3.5) ii) and iii), fij(t)=0 for i=j for any t>0, which shows that F(t,l‘, q) is diagonal with respect to Bq for 76 all t>0. Hence, F ’(t,£‘, q) is also diagonal with respect to Bq. Therefore, Hr Is diagonal with respect to the basis of Tp(Nr) obtained by parallel translating Bq along the geodesic b’: from q to p. Since (Un)p is an element of this basis, U“ is principal at p. Asp is arbitrary in Mr, we are done. // Actually, Proposition 1 is true for any complex space form. The nature of the solution to (3.5) will depend upon the holomorphic sectional curvature of the ambient space and the dimension of the core of thetube, as we will see in the following examples. In the case at hand, i.e. the ambient space being CH“, if N is a totally geodesic submanif old, f ii(t)=sirh(t) or cosh(t) depending upon whether the 11th entry of P is 0 or 1, for i=1,...,2n-2, and f2n-l,2n-l (t)=sinh(2t) or cosh(2t) depending upon whether the coresponding entry of P is 0 or i. 77 Example I: Let N=Hn(-l) be a real space form of constant sectional curvature -l immersed In (2Hn as a totally geodesic and totally real submanif old. (See the proof of Theoreml in [5].) Let Nr be the tube of radius r about N in CH“. If as is a unit normal to N, . at each point p=equ(rt’,)£Nr we can write rp(~,)=1t:,l* : 1,,(10- tttqilnrqul where the Isomorphism is parallel translation AsN is totally real of dill'llZI'lSlOl‘l n Uq=-J£quq(N), so U*peTp(Nr). So let ixI .x 2,...,x ”-1 DC.) be an orthonormal basis of Tq(N). 11 (A,f) is the almost contact structure induced by J onNr, then Bp=txl ”,...,x ,H ”.Afx, ”).....A(x ,H ").u"pl 78 forms an orthonormal basis of Tp(Nr). Setting AXi=e(r,£q)" [A(Xj“)l for each i=1,...,n-i allows us to write Bq=IXI ,...,X H" ,AX‘ ,...,AX “.1 ,Uq} for an orthonormal basis of IEQP. With respect to this basis P, P:l - and REM) have matrix representations: P=diag(l ,H ,0“.I ,1), P*=diag(0n_1,ln., ,0) and R£(t)=diag(-I 2H,-4). for all te(0,r], as endomorphisms on Itqll. Straightforward calculations yield that (3.5) has the matrix solution 79 F (LE, q)=diag(cosh(t)ln.I ,sinh(t)l ".1 ,cosh(2t)). Now, from (3.4), the second fundamental form, Hr, of the tube will have the following matrix representation with respect to the basis (3.6) Hr=diag(tanh(r)l ".., ,coth(r)ln_, ,21anh(2r)). By selecting a suitable frame onN, we see that the representation of Hr depends only upon r and is hence constant on ”r- Nr is obviously contact and in fact satisfies A) «2-4<0. Example 2: Let N=CHk, k=0,l,...,n-l, be a complex space form immersed in CH“(-4) as a totally geodesic submanifold, (see [5]). In case k=0, we are regarding a point to be a trivial complex space form. Otherwise. from [5], the CHk will have constant holomorphic sectional curvature cl 80 -4. Let Nr be the tube of radius r about N in CH“(-4). Let E" be a global unit normal to Nr so that at each p=equ(r£q) we can write r,(~.)= {(6),}: : Tq(N) . [Tq(N)‘n{£ql‘l. AsN is a complex submanif old of CH". Its tangent space Is invariant A under J; so e(r,t’,q)[Tq(N)I Is a A-invariant subspace of Tp(Nr). Let {X1 ,...,X k,JX] ,...,JX k} be an orthonormal basis of Tq(N) and extend to an orthonormal basis Bq={Xl ,...,X k,JX] ,...,JX k’xk‘fl ,...,X ".1 ,JX k.” ,...,JX ".1 ,Uql of {quk The last 2n—l-2k elements of B forma basis of Q Tq(N)1n{£q}1. Due to the invariance under J of Tq(N), we have ¢(xi”)=e(r,tq)Iinl for all i=1,...,n-1. So let Bl Bp={Xl ”,...,X [,AXI “,...,AX k" it I it it I xk,I .....x M .Axm ,...Ax ”.1 .(u )p} be the orthonormal basis of Tp(Nr) obtained by parallel translation of Bq. With respect to B P, P1 and RC“) have the following matrix q. representations: F"(“390 21oO 2h-1-2k ). P‘=diag(02k.12n-j-2k ) and RE(t)=diag(-12n.2,-4) for all t>0. The radially symmetric matrix solution to (3.5), with respect to Bq, has matrix representation 82 F (LE, q)=diag(cosh(t)l 2k,sinh(r)l 2n_zk_2,sinh(2r)). Computing (3.4) for this case yields the following matrix representation for the second fundamental form on Nr with respect to Bp: (3.7) Hr=diag(tanh(r)l 2k,coth(r)l 2n-2k_2,2coth(2r)). In the context of section 2, the geodesic hypersphere (k=0) and the tube around a maximal complex space form (k=n-l) are contact hypersurf aces and satisfy (2.4). These are our models for totally U-umbillic hypersurf aces of CH". The remaining cases k=l,...,n-2 satisfy (2.4), but are not contact. Example 3: Each of the previous examples, despite their obvious differences as tubes with different cores, do haveone thing in common If the 83 bases Bp, p=equ(r£q), are chosen for each r>0 to be compatible (via 2:) with Bq, then we can discuss the limit of the tensor Hr, viewed as acting on span(Bq), as r-too. Clearly, this matrix limit is given by (3.8) lim Hr = diag(l 2n-2:2)° r-ioo The geometric significance of this is obscure from our view of these tensors as acting on the tangent space of the core of the tube. However, in section 6 (originally in [16]), a hypersurface is constructed in a very abstract way that has a second fundamental form of the form (3.8). So we know that such a hypersurf ace exists, the question is whether we can obtain a more geometric characterization In this example, we shall construct a model that can be thought of as a limit of expanding geodesic hyperspheres, called the horosphere, that also has second fundamental form with representation (3.8). B4 Choosea point ptCHn and any direction ££T(CH”). For each r>0, let q(r)=expp(rt',p) and 21': be geodesic with initial direction {p that joins p to q(r). Then p is on each geodesic hypersphere, Gr(q(r)),' centered at q(r) with radius r. It Is known that as q(r) recedes from p (r-m) the Gr(q(r)) approacha limiting hypersurf ace. M°°, called the horosphere(see I7] and [10]). r-No expanding I geodesic j hypersph ares .1 Grtqtr» '7 q(r) \\__ ..z p horosphere Furthermore, the horosphere will have an extrinsic geometry that is obtained as a limiting hypersurf ace of these expanding geodesic hyperspheres. That is, M°° will have a second fundamental form with a representation (3.8) with respect to a suitable basis of BS TD(M°°). This last point will be proven analytically in section 6, when subsequent to considerably more theory, we will be able to show the convergence of the geodesic hyperspheres to a hypersurf ace with a second fundamental form of the type (3.8). Example 4: In this example a relatively simple hypersurface is constructed that has U principal, but no principal curvature is constant. Consider a two dimensional real hyperbolic space form N=H2(-l) immersed in CH3 as a totally real, totally geodesic submanifold. Let Nr be the tube of radius r about N. Then by Proposition 1, Nr is a hypersurf ace that has the direction U as a principal direction For each qu, let Sq(r)={xeNr I d(x.q)=r) denote a cross section of Nr over q. Let {X, ,X 2}q be an orthonormal basis of the tangent plane Tq(N) that extends to an orthonormal basis {XI ,X 2,ij ,JX 2,Y,JY} of T(CH3)| N- If E," is a unit normal field to Nr- then for any peSq(r) we 86 can write (C*)p=8| (oxe. )”+a2(p)ux2)"+n(p)v ”+ctpxm" where the values al , a2, b and c are C°° functions onSq(r) that satisfy l=a I z+a22+b2+c2 at any point peSq(r). Two quick computations will convince the reader that the principal curvatures are not constant on5q(r). Select p such that al =l and a2=b=c=0. Then up has principal curvature 2tanh(2r) and the other principal curvatures at p are tam(r) of multiplicity one and coth(r) of multiplicity three. However, if p is chosen so that a] =a2=c=0 and b=l, then Up has principal curvature 2coth(2r) and the other other principal curvatures at p are tam(r) and 87 coth(r), each of multiplicity two. In fact it can be shown on Sq(r) that Hr is a linear combination (with coefficients the functions a‘, a2, b and c) of four linearly independent diagonal matrices that have their non-zero entries drawn from the set {tam(r),coth(r),2tanh(2r),2coth(2r)}. The conjecture is now that the hypersurf aces of examples 1, 2 and 3 are the only hypersurf aces that are contact or satisfy (2.4). It turns out that this is indeed the case, but this cannotbe shownby viewing CHn only as a complex space form and applying the techniques of sections 1 and 2. To make further progress CHn must be thought of as the base manifold of a certain Lorentzian 51 -fiber bundle. 4. The Lorentzian Circle Bundle over CHn(-4) and its SI -invariant Hypersurf aces The best understood of all non-Euclidean complex space forms, complex projective space, (usually written CP"), is constructed using a natural equivalence on an odd dimensional sphere. 52””. itself immersed in Ch". Complex hyperbolic space can be constructed in a similar way (see [4], and [12], vol Ii). in this case CHn is formed by taking the equivalence on a Lorentzian hyperbolic space form in CM . instead of on a real Riemannian space form. Define a hermitian bilinear form F on C“*1 by n F(z,w)=-20WO+Z zjv'vj i=1 for all z=(zo,zl ,...,z n) and w=(w 0,w, ,...,w n) in CM. F forms a complex Lorentzian metric so that (Iml with the metric F forms a complex Lorentzian space CF" . Cn+1 can also be regarded as a real semi-Riemamian euclidean space, RZZWZ, if it is equipped with the metric Fie(F). Anti-De Sitter spacetime is the hyperquadric defined 89 by it?“ ={zecm‘ |F(z,z)=-l}. The tangent space of anti-De Sitter space is determined by its immersion into the ambient complex euclidean space; explicitly: Tz( H12“ )={we(:n+1 |Re[F(z,w)]=0} for any zeHIZW" . As a real hypersurf ace of 822"”, H13“ has Re(F) | H'Zn'oi as a natural Lorentzian metric that is of constant sectional curvature -l. An S1 ~action can be defined on H1?“ (in fact on (I‘M ) by z-mzf or any ZEH'ZWI and MC with l). | =l. At each point 26mm”, the vector v=iz is tangent to the flow and has length -l. Given a ‘ point 25H,2M , the flow of v through 2 in HEW will be given by the orbit 90 Oz={xt=eit z I teR} that satisfies the differential equation dxt/dt=ix t, which in turn shows that 02 lies in the intersection of the negative definite plane span{2,V} with HP“. Let ~ be the equivalence given by the orbits of the action, i.e. w~z if weOz Then the natural projection 1r: H120“ 4 szm' /~= CHn is a Riemamian submersion with fundamental tensor the natural complex structure J on c."*' , (see [21].). and with time-like totally geodesic fibers, each of which is a trajectory of the vertical vector V=iz at any point 25 H12” I . The complex Riemannian space, CH”. obtained in this way has its complex structure induced from that on CW” , and has constant holomorphic sectional curvature -4, with the metric induced from Re(F). Then the differential of the submersion, m, is a linear isometry ([2il); i.e. it preserves the metric tensor on 91 the horizontal distribution as it projects onto T(CH”). So we shall make nodistinction between the metric on H12n+1 and that on CH". We can now write TZ(H,2“*‘ ) S Tfl(z)( Cl-ln ) o span{v} where the isomorphism is given by m. 0 ll span{V}- 11... as a linear isometry of the distribution h’ of horizontal vector fields on szn" onto T(CH"), induces relations between the connectionsV and V of szm' and CH”, respectively. (See [8] and [21].) These are: (4.1) V(x~) Y~:(va)~+V (4.2) vvtx”i=v(x~,v=(.lx)“'=.l(x"') 92 for all X,Ye T(CH“)E h’, and where X'” or (X)"’ denotes the unique horizontal lift of a vector field X£T(CHn). if ”2"“ is a real hypersurf ace of CH“(-4) then the hypersurf ace F1,2"=11" (ii) of H.2m' , is invariant under the S'-action. and rrlfi:l"1-ir1 is a Riemamian submersion with timelike totally geodesic fibers. Conversely, if F13” is an S'-invariant hypersurface of szn" i, then If la is a Riemamian submersion of Ft onto "2"" =11071) with timelike totally geodesic fibers. Hence, we have the following commutative diagram of immersions and submersions: [71‘2" =: HIZW‘ (-1) in] f." l 11 i Mer ------- -» cu“(-4) where jzli 2“" 4CH"(-4) and jail, 2"»HIZ‘M {-l) are immersions compatible with the f ibration if a global normal it to l“! is selected, its unique lift, it”, is horizontal and forms a global normal to fl. 93 Henceforth, we will drop this distinction between E and it”. Let ('7 be the metric connection of Ft as a hypersurface in H?“ and it be its second fundamental form. The Gauss and Wiengarten formulae in this case are given by (4.3) vxv=vxv-z and Vx£=-RX for all X,Y£T(Fi). Now combining (Li), (4.l), (4.2) and (4.3) we have the f oilowing relations between it and H: (4.4) i'l(x"')=(Hx)"'-r(><)v for all X£T(l‘l)5{V)‘nT(F1), and (4.5) Ftv =U‘. Let Fl be the curvature tensor of Ft. We shall have the opportunity 94 to use the Gauss and Codazzi equations for Fl in H3“: (4.6) fi(x,v)z =Y—><+< i'lv,z>i'lx-iiv (4.7) WXFlW=Wme for all x,VeT(F1). These and the preceeding formulae can be used to prove the f oiiowing useful identities: (4.8) (\‘iwl‘ixxyn =<(vwil)v.2>le-<(v Wi"l)x,z>Fh/ +(t"7 Wi'l)><-(i‘7 wFiW for all x,v,z,w:r(l‘i). and (4.9) (v(x~,Fl)v=l(¢H-lltj)xl"' for any XeT(l’l). 95 Let {XI ,...,X 2n-i } be a frame oni‘l consisting of principal directions in T(M) with corresponding principal curvatures {)‘l ,...,). 2n-l }. Then {Xi “,...,X 2”,] ”,V} forms a frame onF‘l with respect to which Fl is represented by the matrix: i "l f(xi) i O O )‘Zn-l f(xzn-l) l-f(XI ). . . 'f(x2n-i) 0 J. all of whose entries are functions on Fl. Hence, even the stipulation that it be S'-invariant does not guarantee an easy analysis of the structure of it via an investigation of those subbundles of T(Fi) held invariant by Fl, for we are not even assured the existence of principal directions that have real principal curvatures. However, in case U is a principal direction onl‘i, we can choose U=X2n_1 so that Fl is represented by the matrix: 96 diag(x1,...,)\ 2W2, [o< “I ). l-l 0] As we have seen. there are a number of conditions on ii that forceU to be a principal direction on H. in section 2, we saw this for the conditions (2.1) ¢H+H4>=2p4> (ii is contact), and (2.4) ¢H=H¢. ln [lb], (2.4) is shown to be equivalent to Fl being parallel on Fl. Thus, (2.4) satisfied on H shows that F1 is parallel, which in turn yields fi locally symmetric, which finally implies that E is semi-symmetric. Symbollically: (2.4)4vl‘izoavii=o=>ii-ii=o. Surprisingly, ajlthe implications are equivalences for 97 2n+i S‘-invariant hypersurf aces of H. , not just the first, which will enable us to contribute a little toward the work began in [i7]. [231-[26] and [281-[30]. In order to begin the proof of this fact, we must see that when an S‘-invariant hypersurf ace 5% is semi-symmetric its submersion l1=rr(Fl) is a hypersurf ace that has U as a principal direction Lemmajz Let Fl be an S‘-invariant hypersurf ace of szn” and l1=11'(F1). Then fi-FFO on Fl: U is principal on H. Proof: Let {XI ,...,X 2”,, } be a basis of principal directions on it for T(l‘i) with corresponding principal curvatures {in ,...,X 2”,, }. Adirect computation shows that at any point pefi (4m) (iitx,~,xj~)-it)(x,,~.v)v=t(xj)rtxk)(x,2-x,xj+l)x i“ _ . .2- . ."' f(><')f()2, we have xkz-xkx.+l=o, by setting i=l, i=2 then switching j with k in (4.i0) and reading off the second term in the right hand side. This shows that xkzo when k>2. So, after switching the indices back, from the third term in the right hand side of (4.10), we have xk(x,-x2)=o, i.e. x.=x2=x. if there is a third direction, say X3, not in ker(f), then again from the right hand side of (4.i0), we must have x.2->.,x2+l=o, which is impossible in light of $53.2. Hence, Uespan{X,,X2}cD,\, an absurdity. Therefore, U must be principal on H. // 99 For the remainder of this section we will assume that Fl is an S‘-invariant hypersurface of szm' that satisfies the condition "43:0. Let l'l=11'(Fi) and {XI ,...,X 2n-2'U} be a local orthonormal basis of T(M) consisting entirely of principal directions that have corresponding principal curvatures {kl ,...,). 2n.2,ot}. In order to achieve a classification of the S'-lnvarlant hypersurfaces that satisfy this condition, we will need to obtain algebraic consequences for the curvatures associated to the spacelike distribution on F1 that is invariant under Fl, namely: span{X, ”,...,X 2n- 2~}= l/n(ker(f))"'. Lemma 7 if Fl-Fi=0 on an S'-invariant hypersurface Fl of H12“ . let xi). j and xk be principal curvatures on Fl with distinct spacelike principal directions xi“,xj“' and xk”eT(Fl). Then (xixj-lXx i-xj)xk=o. l00 Proof: By direct computation using (4.6) (fitxf'x j”).h)(xi"',xk“')xj"' :[(l‘)\ ilj )(I'X iKk)+O\i)\j")(l’)\ jxknxkfl' so that (H. ixjxl-x ix,,)=(l-x ixjxl-x jxk). if H. iszo then we must have M. ixkzl-x jxk: that is xk(xi->.j)=o. // From Lemma 7, if there is a nonzero principal curvature M with direction Xke h’, then for curvatures xi and )‘j with distinct spacelike directions Xi and Xj different from Xk either a) hat: or b) xi=llx l i- if case 3) holds, then every basis direction different from Xk has the 101 same principal curvature and a simple argument similar to the one employed in the proof of Lemma 6 yields all the nonzero principal curvatures the same value. if case b) is true, all the principal curvatures that have spacelike principal directions must be nonzero and once a nonzerocurvature it has been given, only the values 9. and 1/). are allowable as principal curvatures. Hence. we have only four possible matrix representations with respect to the orthonormal basis {X1 ,...,X 2n-2'U} for the second fundamental form of a real “QPBFSUFfaClZ. M=1r(Fi), of CH", where Fl-Fi=0 on F1: I) H=diag(0 2n-2,0<), ii) H=diag()tl 2n-2,o<), iii) H=diag()\,0 2n-3,0(), or iv) H=diag(M p,(I/)t)l 2n-2-p'°‘) 102 where 9.10 and the basis may need to be reordered for case iii). (Notice that in each case, U" is not principal on F1.) Cases i) and iii) are ruled out by the following lemma: Lemmafi if x is a principal curvature ona semi-symmetric S'-invariant hypersurf ace H of H13” that has a spacelike principal direction, then xz-ax+l=o. Proof: Let X be a spacelike principal direction of A. Then (htx,u ~)-ii)(><,V)U ”=(x2-oa+l)v. // in particular, we see that noprincipal curvature can be zeroif it has a spacelike principal direction Furthermore, the principal curvature 0< must satisfy 0:2-420 on 11. However, an analysis of the values A can attain cannot yet proceed as in section 2 as we do not know if O< and )t are constant. As far as we are concerned the 103 principal curvatures 0< and )t are merely functions on it that satisfy (2.5) xZ-aoulzo. Lmj if Fl-F§=0 onF1 then the principal curvature at with direction U is constant on ii. Proof: The proof is similar to that of Lemma 4. From the proof of Lemma 1, we have Xo<=0 for all Xeker(f). Hence, it is sufficient to show that Uo<=0 on 11. Following the same steps as in the proof of Lemma 4, we recall the equation (2.3) x(uo:)r(v)-v(uo<)i(x)=(uo.)<(4>ll+Hj)v,x>. Again replacing X by U and then Y by U and then substituting the results back into (2.3) yields 104 o=(ua)<(H+H¢j>)v,x> for all X and Y in T(M). Suppose that 0:2-4>0 on an open set U in 11. (Otherwise, O< is a constant :2 on it. ) Then by (2.5) we have the curvature 7t satisfying $221 on U. Choose YernkerU). From the proof of lemma 2, o=(u«)<(xm;z)¢v,x>. 29cm By choosing X=4>Y we have Uo<=0 or 0: x?” = (22.2-00.+o<)\-2)/(2A-o<)= 2(7t2-l)/(2x-o<). A-a The latter equation is ruled out by hypothesis on U, hence Uot=0 on any open subset of l1 with «2-4>0. Therefore, a: is constant on l1. // We now know that an S'-invariant hypersurface F1 of szm' on 105 which fit-13:0. admits a frame with respect to which its second fundamental form has the following possible matrix representations: i) Fl=diag(xl 2H, [0. 1]), or H 0] ii) Fl=diag(xlp.(l/x)l 2W2-p.[o< 1]) H 01 and the second fundamental form of its submersion will have the corresponding representations with respect to a suitable basis: I) H=diag()t12n_2,o<), or II) H=diag()\lp,(l/A)l 2n_z_p,0() where at and 1t satisfy (2.5) and 03-420. Case 1) is obviously a subcase of ii) with p=0 or 2n-2. As in section 2, if «2-4>0, we can set «=2coth(2r) and x=tanh(r) or coth(r). Of course if o<=.+.2 we may set k=zl=1lk Assume that H=diag()tlp,(l/>t)l 2n-2-pv°‘) with respect to a 106 suitably selected orthonormal basis of principal directions. There are two cases to consider: 1. Suppose that p is odd. Then there exists XeD)‘ with ¢X£Dl/)‘. From the proof of Lemma 1 L = 932:2 A 21"“ which implies that Ozax2-4xm. Using (2.5) o=o.(o.x-l)-4x+o.=x(o.2-4) which shows that either x=o or ot2-4=0, neither of which hold by hypothesis. This only leaves the single case: ii. p is even. 107 From the above argument we must have D)‘, and hence Dl/At invariant under 4». Thus, ¢H=H¢ onli whenever its lift is semi-symmetric in Him”. Remark: As the only models we have for S'-invariant hypersurf aces satisfying Fl-B=0 are lifts of certain tubes, one might conjecture that the lift of any tube would satisfy this condition However, a tube of radius r>0 about a totally geodesic real hyperbolic space form (example i) has o<=2tanh(2r) and x=tanh(r) or coth(r) which do not satisfy (2.5). Thus, fi-l‘izo for the lift of this tube. incorporating the preceeding discussion with known results in this area, including those in [4] and [16], yields the following: SI 108 111mm} Let 11,2" be an S‘-invariant hypersurf ace of szr‘" (-1) and i12n'1=r((F1) be its submersion in CHn(-4). Then the following statements are equivalent: 1) 0Fl=0 on Fl. 2) 114:0 on F1. 3) +H=H4> on M. 4) H is cyclic parallel on 11. Proof: 1) => 2) is obvious. 2) =1 3) follows from the preceeding discussion 3) 91) is in [16]. 4) e9 1) is in [4]. // Aclassification of semi-symmetric S‘-invariant hypersurfaces is now possible and will be accomplished in the next section I' 1 109 5. Congruence and Classifications Submanif old theory, as with other branches of mathematics, has its own notion of equivalence. For instance, two hyperspheres of the same radius with different centers in a euclidean space are different merely by location yet obviously have the same extrinsic geometry and as such are extrinsically equivalent submanifolds of the ambient euclidean space. We would say that these spheres are my; and would see the extrinsic equivalence by observing that one sphere can be mapped lsometrically onto the other by making a rigid motion of the ambient space. in this case, a rigid motion of a euclidean space is a translation or a rotation or any combination thereof. (We shall exclude reflections from rigid motions in this study as these reverse orientation) We generalize to semi-Riemamian manifolds: 1 10 Definition: Let F1 be a semi-Riemannian manifold with semi-Riemannian submanifolds l1 and N. 11 and N are congruent if there exists an isometry Q of 11 such that Q I ,1 is an isometry of 11 onto N. We have seen that for submanifolds of euclidean space, congruences are given by translations and rotation However, in the ambient space we have been studying, CH“(-4), the characterization of congruence is not so simple. The principal S'-fiber bundle over CHn will be taken into account as well as the fiber bundle's own imbedding in CF" , i.e. from section 4: H1211” .., C'WI UT CH” We see that rigid motions of (2.”I will induce rigid motions of CH”. Recall that the bilinear form F defined in section 4 forms a 1 11 semi-Riemannian metric on Cn+1 that turns the complex euclidean space into a complex Lorentzian space, CIM , where at the origin (20,0 ..... 0) is a timelike vector if 2020. The bilinear form Re(F) on cn+i , forms a semi-Riemannian metric that turns the complex euclidean space into the real semi-Riemamian euclidean space “22m2’ where at the origin (1,0,...,0) and (i,0,...,0) form a basis of. the negative definite subspace of T0(R22n+2). The isometries of CF” are precisely the group U(l,n)={A£GL(n+1; C): F(Az,Aw)=F(z,w) Vz,w£C,"+' 1. Furthermore, U(1,n) holds len” invariant and acts transitively on H12“. Hence, the elements of U(1,n) will induce isometries of CHn via 11'. Using these ideas, we shall see that isoparametric hypersurf aces of CHn that have Uprincipal and the ”same” second fundamental form are congruent. Consequently, we will obtain nice geometric characterizations of contact hypersurf aces and hypersurf aces satisfying (2.4) in CHn as well as a characterization 112 of semi-symmetric S‘-invariant hypersurfaces of szm. Let fl and N be isoparametric hypersurf aces of CHn that each has the distinguished direction of the induced almost contact structure as a principal direction (Henceforth, we shall write Ufor this direction indiscriminately onl1 and N. Hopefully, the domain of U will remain clear in context.) Suppose that fl and N have second fundamental forms with the same matrix representation with respect to suitably chosen local orthonormal bases of principal directions of T01) and T (N). Let F14) and 11.0 be the simply connected covering spaces of the lifts F1=1r"' (l1) and NWT" (N), respectively. if izl1-t CHn and ij-t CHn are the isometric immersions in complex hyperbolic space and i:F1-t H.2M and j:N-+ szm' are the induced immersions of the respective lifts, we have the following commutative diagram of immersions, submersions and covering maps: 113 F1 N 19 lo 1’ 1 F1 ‘* “12“” ‘- F1 1 "lfi 111' 1 Tim The maps iop:l°1-t H.2"+1 and food)» szm' are now isometric immersions of simply connected Lorentzian spaces of codimensionl into the Lorentzian symmetric space szn‘I , that have the same constant matrix representation f or their second fundamental forms with respect to cannonically chosen orthonormal bases of T01) and T01). 114 Lemming Let N and 11 be real hypersurf aces of CH“(-4) that have: i) Uas a principal direction and ii) the same constant matrix representation for the second fundamental form with respect to suitably chosen bases of principal directions. if N and F1 are the simply connected covering spaces of Nut" (N) and F1=11"1 (r1), then N and F1 are isometric. Proof: We shall use the notation of the preceeding paragraph and diagram. Let xeN and yeFi. By hypothesis and using the ideas of sections 1 and 4 we can choose local orthonormal 138525 ixI ,...,x ”.1 ,qx, ,...,4>x M ,U} of T"(d(x))(N) and (Y, ..... Y ,H .111, ..... 4w IMu} of r"(p(g))(r1) 115 that consist entirely of principal directions, with respect to which the second f undamentai forms have the same representations. (We are allowing 0 to denote the almost contact structure of both N and 11. The domain of 4) will, hopefully, always be clear in context.) To simplify notation set X"=[(o..)" (X"’)]X for any XeT(N) and similarly set Y"=[(p..)'l (Y ”)19 for any Y£T(i1). Let V denote a unit timelike vector on H12“*1 and V" denote either [(m)" (V | Nnx or [(p..)'I (V 111)]y- depending upon the context. Similarly we shall write U" for either [(01..)"(U'”)]x or [(p.)" (U"')]u depending upon the context. We now have local orthonormal bases BX(N)={(XI )“.....(x M )".(j»x, )‘.....(4>x ".1 )‘.u*.v“} of Tx(N), and 3901140,)" ..... (Y n_,)".(4w.)" ..... (<11! n..)".U",V"} of T901). 116 Define a linear isometry ljlzTX(N)-oT 9(F1) by 1((Xi)")=(vi)" and 1((pxi)")=(¢vi)“ for i=1,...,n-1, ~1((U")(,)=(U")y and 1((V“)x)=(V")y. and extend linearly. Let F1 and F1’ denote the second fundamental forms of the isometric immersions ice and Top, and similarly let Fl and Fl' denote the second fundamental forms of the immersions jand i'. As o and p are local isometries, Flx and Fl'u will agree with Flam and 11m). Applying (4.4) and (4.5) we have that iP(le(2)) =Fl’g(‘1'(2)) for any ZeTx(N). Let Pi and 0' denote the curvature tensors of N and F1. By (4.6) 1(fixtx,vizi =R'gttlx,~lv)tlz for all X,Y,ZeT x(ft). Let V and 0' denote the connect ions onN and F1. Using (4.7) and (4.9), it follows that tit maps the tensor (th to the tensor (V1119. 1 17 Hence by (4.8), ill maps (09)x to (011’) y“ Define the mtn covariant differential VmFl at x. (as on p.125, vol. 1 of [12]) by: (Vh)x(x,v;lv)z =vw(R(X,V)Z)-R(v Wx,v)z--h(x,v wviz for all X,Y,W,ZeT x(N), and then for m=1,2,... and any set of vectors {W' ,...,w m-l }CTX(N), SCI (VmN)x(X,Y;W ‘;...;W m-l ;W )Z =thtv m" Fi)x(x,v;w ,;...;w "H )2) -(V"‘" hixlvwxynr 1;...;\r.l m., )2 -(V "l" ii)x(x,v Wwill ,;...;w "H )z m-l '2 (gm-i N)X(X,V;W 1""3‘1’1-1 :waizwi,l ;...;W m-l )2. i=1 Assume that ill maps the tensor (0"1'111)x to the tensor (gm-l My. 118 Then in order to have an inductive proof that til will map (VmN)x to (V’mR')g for any m, it is sufficient to show that F(VXV):V'(11X)('FY) for all X,YeTx(N). But this is indeed the case as N and N are isoparametric and it preserves the metric. Now by Corollary 7.3, Chap. Vl , Vol. 1 of [12], there is a unique isometry QzN-oFl such that (dQ)x=lil. // The point of Lemma 10 is to get into position to establish a general congruence theorem for real hypersurf aces of complex space forms. it is well known that isometric submanifolds of a real space form that have the same second fundamental form are congruent. So congruence results for submanifolds of complex space forms can be established by using known congruence results f or the real space forms that are principal circle bundles over complex space forms. 119 1022mm Let l1 and N be real hypersurf aces of CH“(-4) that each have U as a principal direction and all the principal curvatures are constant. lf l1 and N have the same second fundamental form, i.e. the corresponding principal curvatures are the same, then 11 and N are congruent. Proof: Let tit and Q be as in the proof of lemma 10. Our commutative diagram of the immersions, submersions and covering maps, with the addition of the isometry Q, is now: Q F1 «- fl 1p 10 i i F1 "’ 11'2”” 1" fl 1 n |fi l 11 l 11‘“ I i ll .. cu" «- N As HIZM is a totally umbiilic hyperquadric of the semi-Riemamian manifold Rzzmzzcjm' with second fundamental form -1 on T(HIZ'V'I ), we can chooselocal orthonormal bases of T(N) and T(F1) as 120 in lemma 10 with respect to which the two second fundamental forms have the same constant matrix representations. Let xeN and y=Q(x)eF1. We can identify on” : Tj'(d(x))( 11.2”” ). spanl‘j'totx)» as j(o(x)) and i(p(y)) are unit normals to. H.271” at j(o(x)) and 'i'(p(y)). respectively. Choosefi and f,’ to be unit normals to N and Fl in szm' and write T1(o(x))( ”3”" i 5 TN" ' 593"“ x’ 5 «mm» "61000” )~ . SDMKU ~10“) ’VO(X) a: 000} and 121 q 2 1 2 A I “(904” ( H1 11" 1 - T (1'1) 0 808111: 9} 9 Hence, we can regard F1 and N as Lorentzian submanifolds of the semi-Riemannian manifold 822””. Let at be a curve in N beginning at x and denote normal parallel translation of vector fields along the curves o< and Q(o<) by P,x and PQ(o<) respectively. As H121“ is a totally umbillic hyperquadric of 822’”, parallel translation in szm' preserves the second fundamental form of HIZM. Define 11*:TX(N)1-o T901)1 by lll‘(x)=y and F‘(EXFE'U and then extend linearly to obtain a linear isometry of the two dimensional normal spaces. A linear isometry ‘1'1'0((5)IT°((5)(~)1"TQ(«(S» (F1)Jl can be defined by setting ‘Flo<(s)=PQ(o<(s))° lploP°((s)". At each s in the domain of 0(, (0013494045) is an isometry that maps the second fundamental form of N at «(5) onto that of F1 at mods». Therefore. the linear isometry (11°45) will map the second fundamental form of ‘1 122 N in CF” at «(5) to that of F1 in Cjn” at mods», i.e. 1"“($)B(X.Y)=B’( (dQ)o((5)X. (00)“(5)Y) where vxv=vst(x,v) for all X,YeT “(5(0), and vxv=v'st'(x,v) for all X,YeT 9(a(s))(l°1) are the Gauss formulae of N and F1 in CF” . (in fact, BX(X,Y)=< fix»: x-x for all x,v:r "(it) with the analogous statement for Tg(F1).) Now by Theorem 41, chap. 4 of [201, there exists an isometry .7. 123 of Cf“ such that 9| ~=Q 1? induces a rigid motion of CHn that maps N lsometrically to i'l.// With Theorem 3 we can classify the hypersurf aces of section 2 in terms of the examples of section 3. Iheorem 4 Let (1 be a complete connected contact hypersurf ace of CH"(-4). n23. Then 11 is congruent to one of the following: i) A tube of radius r>0 around a totally geodesic, totally real hyperbolic space form H"(-1), ii) A tube of radius r>0 around a totally geodesic complex hyperbolic space form CHn'l (-4), iii) A geodesic hypersphere of radius r>0, or iv) A horosphere. Pr00f: The proof of cases i)-iii) is obvious. An analytic proof of the existence of a horosphere in CH"(-4) is postponed until the next 124 section // Theorem 5 Let 11 be a complete connected real hypersurf ace of CH"(-4) that satisfies (2.4) ¢H=H¢x Then 11 is congruent to one of the following: i) A tube of radius r>0 around a totally geodesic CHp(-4), Ospsn-l, or ii) A horosphere. Proof: Again 1) and ii) are obvious if we grant the existence of a horosphere in CH"(-4). // in section 4, semi-symmetric hypersurf aces that are also S'-invariant are characterized as lifts of real hypersurfaces in CHn 125 that satisfy (2.4). Hence: r r 1 Let 11,2“ be a semi-symmetric hypersurf ace of Him” that is S'-invariant. Then 11.2” is congruent to an S'-fiber bundle over either a tube about a complex hyperbolic space CHp(-4), p=0,l,...,n-l, imbedded as a totally geodesic complex submanifold of CH'Y-4), or a horosphere. // 126 6. An Analytic Construction of a Horosphere Using the congruence results of section 5, we can now place the model spaces used in [16] into the context of the preceeding geometric classification First, recall an elegant and well-known , (eg. [16] and [18]), analytic method of determining the extrinsic geometry of a level hypersurf ace of a C°° function ona space form imbedded as a hypersurf ace in a euclidean space, modified to fit the particular needs of this section Let f: 82”" 48 be a C°° function and l1,"(c) be an imbedded Lorentzian space form in 112"" of sectional curvature c. Let Vf denote the gradient of f in T01) as a function on 11, and 0f denote the gradient of f as a function on 82"". Let S be the set of all 568 such that F15=f '1 (s) is a hypersurf ace of 82"" . Then for any 553, F1,5 has Vf/Wfl as a unit normal f ieid in T(RZ'M ). Similarly, let T be the set of all 568 such that Ms=fisnl1'”(c) is a hypersurf ace of 11.“(c). Then 115 will have Vf/lVfl as a unit normal in T( ll,“(c)), for each 127 seT. For a given f, 0f is usually easy to calculate. Once this is done, (60 vr=f7t+c where <, > is the standard metric of 82M and t, is a unit timelike . normal field to l1.“(c) in 82"". (Notice that the choice of positive coefficient of C is necessitated by the causal nature of I). Let Hess(f;Hzn" ) denote the hessian of f as an operator on 82"” . For each 565, the second fundamental form Fl of Fls in F12”I is given by (6.2) 41x.» =Hess(f: 82"” )(x.v)/| tin for all X,YeT(Fl$) which for a given f is usually easy to compute. For each seT. the second fundamental form H of M5 in 1'1.”(c) can be obtained similarly: 128 (6.3) =Hess(t:rl.“(c) )(x,v)/lv fl for all X,YeT( 115) and where Hess(f;l11“(c)) denotes the hessian of f as an operator on T(M,”(c)). Once (6.2) has been computed, (6.1) can be used to obtain a representation of (6.3), thereby yielding an explicit calculation of the second fundamental form of "s in 11.n(c). Example; Consider the function GD:C,n+1 48, for each p=0,1,...,n defined by D Gp(2)=-i20i2+2 I4 I2 i=1 where z=(zo.zl ..... z n). For r>0. define a level hypersurf ace of Cf“ by Flp(r)={26 CF" |Gp(z)=-cosh2(r)}. n+1 The gradient of GI) in C. is computed to be 129 VGp(z)=2(zo,zI ,...,z p,0,...,0) for all 26 Cjn" , so that 0Gp(z)/2cosh(r)=(sech(r)z),sech(r)zl ,...,sech(r)z p,0,...,0) is a unit (time-like) normal to Flp(r) in CF" , for all zeFlp(r). T112 121/21 hypersurface Mp(r)=Flp(r)n HIZM is nothing more than the model 112p” 2q+1 (tam2(r)) of example 4.1 in [16]. Notice that lip(r) is isometric to the product H129" (-cosh2(r))x32("’pH (sinh2(r)). The gradient of Gp on 11p(r) in T(H.2"" ) is given by (6.1): 130 VGp(z)=-2cosh2(r)(tanh?(r)zo,...,tanh2(r)z ptzp+1 ,...,z n) for all 25 11p(r). Thus, (1(2):-(tanh(r)zo,...,tam(r)z p,coth(r)zp,, ,...,coth(r)z n) is a unit normal to Mp(r) in mm" , for all zenp(r). At this point we can see that the second fundamental form of 11D(r) in HIZM is diagonalizable with respect to a real basis of Tz(r1p(r))={z,i:(z)}l and has constant principal curvatures tanh(r) and coth(r) of real multiplicities 2p+l and 2n-2p-1, respectively. Let Np(r)=n(rb(r)) and U denote the distinguished vector on Np(r) viewed as a real hypersurf ace of CH”. Let H' and H denote the second fundamental forms of 11p(r) and Np(r), respectively. We can write Ufi(zf-J(flu(£(2)))=m(-i£(2)). i.e. (mi-tau). 131 An explicit calculation of H’(U"’)Z using (6.3) followed by an application of (4.4) shows that Ufl(z)is principal in T“(Z(Np(r)) with curvature 2coth(2r). Subsequent calculations yield the other principal curvatures tanh(r) and coth(r) of multiplicities 2p and 2n-2p-2, respectively, each of which having a (1)-invariant eigendistribution From the work of sections 3 and 5 we see that ' Np(r) is congruent to a tube of radius r about a totally geodesic complex space form in CHn isometric to a CH”. in particular, we shall need that N0(r) is congruent to a geodesic hypersphere of radius 1'. 5320112126 Consider the function Gzcjn“I -tFl given by G(z)= | 20-2, P, where z=(zo....,z n)- and the level hypersurf ace of CF” , F1=G’I (l). The gradient of G in CF" canbe written as 132 V7G(z)=2(zl -zo,z‘ -zo,0,...,0) for all 25 F1. The level hypersurf ace N=F1nH,2”"' is the model hypersurf ace N of example 4.2 of [16]. The gradient of G in HIZM is given by VG(z)=2(z].221-zo.22 ..... z n) for all ZEN. H211C2, £(z)=(zl .221 -zo,22,...,z n) is a unit normal to N in H.211" . Set 11n*=11(N),as in example 4.2 of [16]. Explicit calculations using (6.3) and (4.4) show that U is principal on 11"” with curvature 2, and that 1 is a principal curvature of multiplicity 2n-2, i.e. the 133 second fundamental form of "n“ acts as the identity transformation on ker(f). Thus, Mn,‘ is our candidate for a horosphere. in order to see this and thereby complete the classification analytically we will show that "n” is a limiting hypersurf ace of a specific family of geodesic hyperspheres. Other than the fact that Fin" and a horosphere have the same second fundamental form, it is not clear that "n” can be realized as the limiting hypersurface of a certain family of expanding geodesic hyperspheres. in the f oilowing discussion we shall use the hypersurf aces of Him” constructed in example 5 to show that this is indeed the case. Let P=(l,0,...,0)£H, 2"" and consider the geodesic emanating from 1r(P) in CHn given by 3(r)=1t(cosh(r),sinh(r),0,...,0). (See p. 285 of [12], Vol 11.) As in example 3, each geodesic hypersphere of radius r centered at 21’ (r) contains the point 1r(P). We will see that these hyperspheres convergeto a limiting hypersurf ace, namely 11"”. 134 Earlier in this section we discovered that the hypersurf ace of 1112"” defined by l'lo(r)={zeH,2n+l | tanl12(r) 1201 2:2 n 12112 } i=1 is actually the lift (up to a congruence, of course) of a geodesic hypersphere of radius r. Notice that 3(r)er10(r) and that 1r(P) is equidistant from every point on 110(r). So.1r(P)plays the role of center of 1r(110(r)) in CH”. in particular, we see that the family of hypersurf aces {11'(l“io(r)) I DD) is not our candidate for the convergent family. However, all is not lost, for we should be able to find a rigid movement of CH", induced by an A(r)£U(l,n), that for each r>0 will translate 11(110(r)) to a geodesic hypersphere of radius r and center 3 (r) in such a way that the family {11(A(r)[(l‘10(r))]) I r>0}will converge to a limiting hypersurf ace. This limiting hypersurf ace will be Mn” =11" (N), which must then be a horosphere through 1r(P). For each r>0, let A(r)eU(l,n) be defined by 135 A(r)=diag{ [cosh(r) -sinh(r)1-I n—l }- lsinh(r) -cosh(r)J A(r) is a rigid motion that maps 110(r) onto the lift of the geodesic hypersphere of radius r centered at b’ (r), and therefore induces a rigid motion of CHn that moves the geodesic hypersphere 110100)) that has radius r and center 1r(P) onto the geodesic hypersphere that has radius r and center 21’ (r) and contains 1r(P). r—Mo 7i .1 11(9( 1'11 11.001) ’U I ...... -- - fl( (“1.01) 136 Egepgsition z N is the limiting hypersurface of the family {A(r)[110(r)1| r>0} of hypersurf aces in HF"+1 , i.e. lim {A(r)irio(r)li = N ("-100 and is therefore an S‘-fiber bundle over a horosphere. Proof: For any z=(zo,zl ,...,z n)lsl“l(,(r) we have I 201 2=cosh2(r) and n 2 12112 = 8101120). F" Let w=(w0,w' ,...,w n)£A(r)[l10(r)l. Then 137 w=(cosh(r)zo-sinh(r)z' ,sinh(r)z o-cosh(r)zl ,zz,...,z n) for some 261100). in particular, we have 1W0’Wl |=(cosh(r)-sinh(r))|zo+zl |=e'r|zo+zl I. Thus, Iwo-w, | .<. e'r(|zol+ 12, I): e'r(cosh(r)+sinh(r))=l Which shows that the limiting hypersurf ace llm {A(r)[l10(r)l} must r400 satisfy 120'21 ls l for any z=(zo,...,z n)ls lim {A(r)ll10(r)l}. r-OOO To see the reverse inequality, let Fl>0 be given For each r>Fl, consider the disc S(r.R)={weA(r)illo(r)l I d(P.W)0 r-voo r-roo This establishes the existence of a horosphere analytically in 139 CHn and thereby completes the preceeding classification Notice that the representation of a horosphere as a submersed level hypersurf ace depends both on the choice of PeH,2W1 and on the geodesic emanating from P; equivilantly: upon the choice of normal to the lift of a horosphere at P. it is interesting to note that we obtain different bounds for 12071 [2 fora limiting hypersurf ace of a convergent family of S'-fiber bundles over geodesic hyperspheres if a different geodesic emanating from P is selected. in [2]. the converseto Proposition 1 is proved for the ambient space CP”, which allows a classification of its real hypersurf aces that have the direction U principal. if we enlarge the class of tubes to include horospheres (as hyperspheres of infinite radius and centered at points at inf lnity), i believe the converse to Proposition 1 is also true if the ambient space is CH". 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