THREE-MANIFOLDS OF HIGHER RANK
By
Samuel Zhong-En Lin

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics - Doctor of Philosophy
2017

ABSTRACT
THREE-MANIFOLDS OF HIGHER RANK
By
Samuel Zhong-En Lin
Fixing ε = −1, 0, or 1, a complete Riemannian manifold is said to have higher hyperbolic,
Euclidean, or spherical rank if every geodesic admits a normal parallel field making curvature
ε with the geodesic. In this thesis, we establish rigidity results for three-manifolds of higher
rank without a priori sectional curvature bounds. Complete finite volume three-manifolds
have higher hyperbolic rank if and only if they are finite volume hyperbolic space forms.
Complete three-manifolds have higher spherical rank if and only if they are spherical space
forms.
In addition to the rigidity results, we also provide constructions of non-homogeneous
manifolds of higher hyperbolic rank of infinite volume. These examples show the necessity
of the finite volume assumption in hyperbolic rank rigidity results.

ACKNOWLEDGMENTS

First of all, I would like to thank my thesis advisor Benjamin Schmidt for introducing this
problem to me, and furthermore, for his guidance and encouragements during these years. I
am extremely grateful for the long hours he spent to meet with me, in which he listened to
me with patience and explained mathematical knowledge and ideas with clarity.
I would also like to thank the faculty and student members in the Geometry and Topology group here at Michigan State University for creating such an enjoyable and supportive
research environment. I would like to thank Professor Wolfgang Ziller for his interest and
comments on this work.
Finally, I would like to thank my parents for their tremendous support. This dissertation
is dedicated to them.

iii

TABLE OF CONTENTS

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 A Partition of Three-manifolds of Higher Rank . . . . . . . . . . . . . . . . .
2.2 Manifolds of Extremal Curvature . . . . . . . . . . . . . . . . . . . . . . . .

7
8
11

Chapter 3
3.1 Local
3.1.1
3.1.2
3.2 Local

Local Structures . . . . . . .
Structures on Generic Sets . .
Pointwise Calculations . . . .
Calculations Along Geodesics
Structures on Extremal Sets . .

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Chapter 4 From Local to Global . . . . . . . . . . .
4.1 Evolution Equations Along Isocurved Geodesics in
4.1.1 Existence of Isocurved Geodesics . . . . .
4.1.2 Evolution Equations . . . . . . . . . . . .
4.2 Evolution Equations Along Isocurved Geodesics in

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Extremal Sets

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Chapter 5 Rigidity Theorems
5.1 Spherical Rank Rigidity . .
5.2 Hyperbolic Rank Rigidity .
5.3 Euclidean Rank Rigidity . .

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Chapter 6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Warping hyperbolic 1-space over the Euclidean line. . . . . . . . . . . . . . .
6.2 Main Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 7 Concluding Remarks
7.1 Constructing Examples . . . .
7.2 Curvature Assumption . . . .
7.3 Stronger Rank assumption . .

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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1
Introduction
Locally symmetric spaces are defined as spaces symmetric with respect to local geodesic
involutions, or equivalently, spaces with parallel curvature tensors. Properties of locally
symmetric spaces were first studied systematically by Elie Cartan through algebraic classifications. These properties indeed manifest the large symmetries within this special class of
Riemannian manifolds. For example, every geodesic on a certain class of locally symmetric
spaces (compact rank one symmetric spaces) is contained in a totally geodesic two sphere
of constant sectional curvatures one. It is natural to ask if these structures of symmetry
characterize locally symmetric spaces.
Rank rigidity results characterize locally symmetric spaces through the following geometric notions of rank. Fixing ε = −1, 0, or 1, a complete Riemannian manifold is said to have
higher hyperbolic, Euclidean, or spherical rank if every geodesic admits a normal parallel
field making curvature ε with the geodesic.
Manifolds of higher hyperbolic (respectively spherical) rank includes noncompact (respectively compact) rank one symmetric spaces and their Riemannian quotients. Manifolds
of higher Euclidean rank includes Riemannian products of manifolds and locally symmetric
spaces of higher real rank.
Historically, the notion of Euclidean rank first appeared in the study of nonpositively
curved manifolds. Roughly speaking, the Euclidean rank reflects the amount of infinitesimal

1

flat planes in nonpositively curved manifolds. It turns out that structures of nonpositively
curved compact manifolds of higher Euclidean rank are quite rigid. Ballmann [1], and independently by Burns and Spatzier [5], proved that a compact, locally irreducible, nonpositively curved manifold of higher Euclidean rank is locally symmetric. The proof of Ballmann
also works when the compactness assumption is replaced by the finite volume assumption
and a lower bound on sectional curvatures. The Euclidean rank rigidity result was further
generalized by Eberlein and Heber in [9] to nonpositively curved manifolds whose fundamental groups satisfy the duality condition, a dynamical assumption on their actions on the
universal cover.
On the other hand, nonnegatively curved manifolds of higher Euclidean rank are not
rigid. In [17], Spatzier and Strake construct many locally irreducible manifolds of higher
Euclidean rank that are not symmetric.
Hamenst¨adt first generalized the notion of rank into the hyperbolic setting, proving that
compact manifolds of higher hyperbolic rank are locally symmetric if sec ≤ −1 [10]. The
sectional curvature bound was relaxed by Constantine in some cases. He showed that, when
the dimension is odd, or when the sectional curvatures are very closely pinched, a compact,
nonpositively curved manifold of Euclidean rank one (means not having higher Euclidean
rank) has higher hyperbolic rank if and only if it is real hyperbolic [8]. Recently, Connell,
Nguyen, and Spatzier showed that a compact manifold of higher hyperbolic rank is locally
symmetric if its sectional curvatures are bounded between −1 and − 41 [6].
The first spherical rank rigidity result was given by Shankar, Spatzier, and Wilking.
Through a Morse theoretic approach, they showed that manifolds of higher spherical rank
are locally symmetric if sec ≤ 1 [16]. Partial spherical rank rigidity results were obtained by
Schmidt, Shankar, and Spatzier in [14] for manifolds with sec ≥ 1.
2

In contrast, there are fewer rank rigidity results without a priori assumptions on sectional curvatures. Existing results include the works of Molina-Olmos [12], Watkins [18],
and Bettiol-Schmidt [3], where Euclidean rank-rigidity results are proved after replacing
curvature assumptions with assumptions on having many flats, having no focal points, and
dimension, respectively. Specifically, Molina and Olmos showed that a manifold that satisfies
a strong irreducibility condition must be locally symmetric if every geodesic is contained in
a closed immersed flat of dimension at least 2. The work of Watkins implies that a finite
volume, locally irreducible manifold of higher Euclidean rank without focal point is locally
symmetric. Bettiol and Schmidt proved that the universal cover of a complete three-manifold
splits isometrically if every geodesic in the manifold admits a parallel Jacobi field. Notice
that, without a priori assumption on sectional curvature bounds, a parallel field making
curvature 0 with the geodesic needs not to be Jacobi.
The main goal of this thesis is to establish hyperbolic and spherical rank rigidity results
in dimension three without a priori sectional curvature bounds. In other words, we prove
the following two rigidity results.
Theorem 1. A finite volume complete Riemannian three-manifold M has higher hyperbolic
rank if and only if M is a finite volume hyperbolic space form.
Theorem 2. A complete Riemannian three-manifold M has higher spherical rank if and
only if M is a spherical space form.
In the Euclidean rank case, we show that one can replace the Jacobi assumption with the
finite volume assumption in the rank rigidity theorem of Bettiol and Schmidt [3, Theorem
A].

3

Theorem 3. Let M a finite volume three-manifold of higher Euclidean rank. Then the
universal cover of M must split isometrically as N × R.
The finite volume assumption in Theorem 1 is necessary. In this thesis, we give constructions of non-homogeneous manifolds of higher hyperbolic rank with infinite volume. The
following theorem is a cleaner version of Theorem 6.1.
Theorem 4. For each dimension d ≥ 3, there exist a one parameter family of complete,
smooth Riemannian metrics gs on Rd with infinite volume such that
(1) gs are not homogenerous with infinite volume,
(2) every gs geodesic is contained in a totally geodesic hyperbolic plane of curvature −1,
and
(3) the sectional curvatures of gs are between −1 and −2 + es .
Note that (2) in Theorem 4 implies that these examples have higher hyperbolic rank.
To my knowledge, these are the first examples of non-homogeneous manifolds of higher
hyperbolic rank. It shows that infinite volume manifolds of higher hyperbolic rank may even
fail to be homogeneous, even when the sectional curvatures are arbitrary closely pinched. A
more detailed description of properties of these spaces is in Chapter 6.
It should be noted that a non-symmetric manifold with infinite volume and higher hyperbolic rank has already been constructed by Connell in [7]. Homogeneous negatively curved
manifolds split as semidirect products of R and nilpotent Lie groups. The homogeneous
examples was constructed on the space obtained by gluing CH 2 and a real hyperbolic plane
along the R factors. Our last theorem show that these homogeneous examples, however, do
not exist in dimension three.
4

Theorem 5. A homogeneous three-manifold of higher hyperbolic rank is a hyperbolic space
form.
Our strategy for proving the rigidity theorems is to analyze the local structure of threemanifolds of higher rank in terms of Ricci diagonalizing orthonormal frames. This is roughly
done as follows. At non-isotropic points, the Ricci tensor either has two or three distinct
eigenvalues. The set of points where the Ricci tensor has three distinct eigenvalues is called
the generic set. The majority of work in this paper is to show that on the generic set,
the Christoffel symbols of the Ricci diagonalizing frame satisfy an overdetermined system of
differential equations. From the system of equations, we deduce that the generic set must be
empty for manifolds of higher spherical rank, finite volume manifolds of higher hyperbolic
rank, or finite volume three-manifolds of higher Euclidean rank. Hence the rigidity problem
for a three-manifold M of higher rank is reduced to the case when M has extremal curvatures,
that is secp ≥ ε or secp ≤ ε for each p ∈ M .
Three-manifolds with cvc(ε), a pointwise notion of having higher rank, and with extremal curvature ε are studied by Schmidt and Wolfson in [15]. Their structural results are
strengthened in this thesis by the higher rank assumption, leading to the desired hyperbolic
or spherical rigidity results.
In the Euclidean rank case, the parallel fields making curvature 0 with geodesics are
actually Jacobi if M has extremal curvatures. Hence the Euclidean rank rigidity result
directly follows from the rank rigidity result of Bettiol and Schmidt after showing that the
generic set is empty.
We organize the thesis as follows. In Chapter 2, we introduce preliminary results, including a partition of three-manifolds of higher rank into three disjoint sets: isotropic set,
extremal set, and generic set. To establish Theorem 1, 2 and 5, it suffices to show that the
5

generic set and the extremal set are empty. To do so, we first study the local structures of
generic and extremal sets in Christoffel symbols of Ricci diagonalizing orthonormal frames
in Chapter 3. We then derive differential equations of geometric quantities in Chapter 4. In
the last section of Chapter 4, we show that the extremal set is empty for the hyperbolic and
spherical cases provided that the generic set is empty. Using the differential equations derived in Chapter 4, we prove the rigidity theorems in Chapter 5 by showing that the generic
set must be empty. The constructions of non-homogeneous manifolds of higher hyperbolic
rank are in Chapter 6. Finally, we give conjectures and possible directions for future works
in Chapter 7.

6

Chapter 2
Preliminary
This section introduces notations and preliminary results. Most of the results in this chapter
can be found in [15]. Unless otherwise stated, M denotes a complete Riemannian threemanifold with Levi-Civita connection ∇ and curvature tensor R. For each p ∈ M , let Tp M
and Up M denote the tangent space and unit tangent space at p respectively.
As M is three dimensional, an orthonormal frame {e1 , e2 , e3 } of Tp M diagonalizes the
Ricci curvature tensor if and only if

R(e1 , e2 , e3 , e1 ) = R(e1 , e2 , e2 , e3 ) = R(e1 , e3 , e3 , e2 ) = 0.

(2.0.1)

Let λij = sec(ei , ej ). Throughout the remainder of the paper, we assume that indices have
been chosen such that
λ13 ≤ λ12 ≤ λ23 .

(2.0.2)

Direct computations using (2.0.1) shows that all sectional curvatures are computable in terms
of the λij as described in the next lemma.
Lemma 2.1. Let σ be a 2-plane with unit normal vector u = c1 e1 + c2 e2 + c3 e3 . Then
sec(σ) = c21 λ23 + c22 λ13 + c23 λ12 .

7

2.1

A Partition of Three-manifolds of Higher Rank

Fix ε = −1, 0, or 1. Recall that M has higher rank if every geodesic γ : R → M admits
a parallel field V (t) perpendicular to γ (t) such that sec(V (t), γ (t)) ≡ ε for all t ∈ R. The
following pointwise version of the rank assumption was introduced in [15] to study the local
structure of manifolds of higher rank.
Definition 2.2. A Riemannian manifold has constant vector curvature ε, denoted by cvc(ε),
if every tangent vector is contained in a curvature ε plane.
Clearly, manifolds of higher rank have cvc(ε). For cvc(ε)-manifolds, the following argument using Lemma 2.1 and (2.0.2) implies that λ12 equals ε. If λ12 < ε, then λ13 < ε
and the vector e1 is not contained in a curvature ε plane, which is a contradiction. Hence
λ12 ≥ ε. A similar argument shows that λ12 ≤ ε, which implies that λ12 = ε.
Letting λ = λ13 and Λ = λ23 , we then have

λ13 = λ ≤ ε ≤ Λ = λ23 .

(2.1.1)

Definition 2.3. A point p ∈ M is said to be
(1) isotropic if λ = Λ = ε,
(2) extremal if precisely one of λ or Λ equals ε,
(3) generic if λ < ε < Λ.
Throughout, let I, E, O denote the set of isotropic, extremal, and generic points respectively. A three-manifold M of higher rank is the disjoint union

M = I ∪ E ∪ O.
8

As the sectional curvature is constant on I, hyperbolic or spherical rank rigidity occurs when
E = O = ∅.
Note that for each p ∈ E, either λ < ε = Λ or λ = ε < Λ at p. This motivates the
following partition of E.
Definition 2.4. Define subsets E− and E+ of E by

E− = {p ∈ E | λ = ε}

and
E+ = {p ∈ E | Λ = ε}.
When p is not an isotropic point, one can apply Lemma 2.1 to characterize the curvature
ε planes in Tp M in terms of their normal vectors.
Lemma 2.5. Let p ∈ E ∪ O and let σ be a 2-plane in Tp M with unit normal vector u =
c1 e1 + c2 e2 + c3 e3 . Then sec(σ) = ε if and only if
(1) c1 = 0 when p ∈ E− ,
(2) c2 = 0 when p ∈ E+ ,
(3) c1 = ±mc2 when p ∈ O, where m =

ε−λ
Λ−ε .

Proof. Immediate from Lemma 2.1.
We define the following disjoint partition of Up M .
Definition 2.6. A vector X ∈ Up M is
(1) isocurved if every 2-plane in Tp M containing X has curvature ε,
9

(2) unicurved if X is contained in only one curvature ε 2-plane,
(3) generic if it is neither isocurved nor unicurved.
Definition 2.7. A geodesic γ : (k, l) → M is isocurved if γ (t) is isocurved for all t ∈ (k, l).
Since the normal vectors of 2-planes containing X form a two dimensional vector subspace, Lemma 2.5 implies the following characterization of isocurved vectors.
Corollary 2.8. Let p ∈ E ∪ O and let X = x1 e1 + x2 e2 + x3 e3 ∈ Up M . Then X is isocurved
if and only if
(1) x1 = ±1 if p ∈ E− ,
(2) x2 = ±1 if p ∈ E+ ,
(3) x3 = 0 and x2 = ±mx1 if p ∈ O.
On the interior of the generic set or on the interior of the extremal set, it is possible to find
a local orthonormal frame that satisfies (2.0.2). Let {e1 , e2 , e3 } be such a local orthonormal
frame. Throughout, we use the Christoffel symbol Γkij to denote < ∇ei ej , ek >. As the
j

connection is metric, we have that Γkij = −Γik . In particular, Γkik = 0.
Lemma 2.9. For each p ∈ O,
Γ311 = m2 Γ322
Proof. The lemma follows from (2.0.1) and the differential Bianchi identity

(∇e3 )R(e1 , e2 , e2 , e1 ) + (∇e1 )R(e2 , e3 , e2 , e1 ) + (∇e2 )R(e3 , e1 , e2 , e1 ) = 0.

10

(2.1.2)

Remark 2.10. As the eigenvalues of the Ricci tensor are distinct at points in O, the
eigenspaces define three distinct global smooth line fields on O. Given a connected component C of O, each of these line fields may or may not be orientable over C. In the case
where one of these line fields is not orientable on C, it resolves to an oriented line field on
some double cover of C. As the rank assumption and finite volume assumption passes to finite covers, we may always assume that there exists a global Ricci diagonalizing orthonormal
frame {e1 , e2 , e3 } on C (the finite volume assumption is required for the proof of hyberbolic
rank rigidity).

2.2

Manifolds of Extremal Curvature

Definition 2.11. A three-manifold M of cvc(ε) is said to have extremal curvature if for
every p ∈ M , secp ≥ ε or secp ≤ ε.
The definition is equivalent to O = ∅, or equivalently, M = E ∪ I. The following
proposition, a structural result used in subsequent sections, is proved in [11] or [15, Corollary
2.10].
Proposition 2.12. Let M be a three-manifold of cvc(ε) and with extremal curvature. For
each connected component of C of E, there exists a complete geodesic field E on C or a double
cover of C whose integral curves are isocurved geodesics.
The structural result leads to the following rigidity theorems proved in [15, Theorem 1.2]
and [15, Theorem 3.5] respectively. A different proof of Theorem 2.13 is given in Chapter 4.
Theorem 2.13. A finite volume complete Riemannian three-manifold with extremal curvature -1 and higher hyperbolic rank is hyperbolic.
11

Theorem 2.14. Assume that M is a connected, simply-connected, complete and homogeneous three-manifold with extremal curvature -1. If M has higher hyperbolic rank, then M
is isometric to the three dimensional hyperbolic space.

12

Chapter 3
Local Structures

3.1

Local Structures on Generic Sets

This section studies the local structure of generic points in a three-manifold of higher rank.
As the generic set O is open, we may consider a local orthonormal frame {e1 , e2 , e3 } and
satisfying (2.1.1). With respect to this local orthonormal frame,

λ < ε < Λ.

(3.1.1)

Specifically, this section derives local equations satisfied by the Christoffel symbols with
respect to this frame in a three-manifold of higher rank. Roughly speaking, these equations
will be derived as follows.
Lemma 3.2 below will show that each generic vector tangent to O belongs to exactly
two curvature ε planes. Generic vectors are characterized by an open condition on tangent
vectors. Hence, given a geodesic with generic initial velocity vector, its velocity vectors
remain generic for short time, and consequently, also belong to exactly two curvature ε planes.
As geodesics have higher rank, they are also contained in a parallel family of curvature ε
planes. As a result, the parallel field equations apply to at least one of these two families of
curvature ε planes along generic geodesics. Essentially, these parallel field equations imply
the desired Christoffel symbol relations.
13

3.1.1

Pointwise Calculations

Given a generic point p ∈ O, each unit tangent vector X ∈ Up M may be expressed in the
frame above as X = x1 e1 + x2 e2 + x3 e3 , where x21 + x22 + x23 = 1. For simplicity, we will
frequently write X = (x1 , x2 , x3 ), omitting the frame vectors from our notation.
Lemma 3.1. Let p ∈ O. For X = (x1 , x2 , x3 ) ∈ Up M a generic or unicurved unit vector,

1 + m2 − (mx2 − x1 )2 > 0

and
1 + m2 − (−mx2 − x1 )2 > 0.
Proof. We prove the first inequality; the second inequality is obtained analogously after
replacing m by −m. Note that

(mx2 − x1 )2 = |(x1 , x2 ) · (−1, m)|2 ≤ (x21 + x22 )(1 + m2 ) ≤ 1 + m2 ,

(3.1.2)

where the first inequality is from Cauchy-Schwartz and the second inequality is from the
fact that X is a unit vector. The statement of the lemma holds if one of these inequalities
is strict. The first inequality is strict unless x2 = −mx1 , and the second inequality is strict
unless x3 = 0. The lemma now follows from Corollary 2.8.

By Lemma 3.1, we may associate to each generic or unicurved unit vector X = (x1 , x2 , x3 )
the numbers

14

1 + m2 − (mx2 − x1 )2

(3.1.3)

1 + m2 − (−mx2 − x1 )2 ,

(3.1.4)

1
1 + mx1 x2 − x21 , mx22 − x1 x2 − m, (mx2 − x1 )x3
K

(3.1.5)

1
1 − mx1 x2 − x21 , −mx22 − x1 x2 + m, (−mx2 − x1 )x3 .
L

(3.1.6)

K=
and
L=
and the following two unit vectors:

V+ =

and

V− =

Direct computations show that V + , V − ∈ Up M , and V + , V − ⊥ X. Note that the expression
for V − can be obtained by changing every m in the formula for V + to −m.
Lemma 3.2. Let p ∈ O and let X = (x1 , x2 , x3 ) ∈ Up M be a unicurved or generic unit
vector and let V = (v1 , v2 , v3 ) be a unit vector perpendicular to X.
(1) If x3 = 0, then V + = ±V − . Moreover, sec(X, V ) = ε if and only if V = ±V + .
(2) If x3 = 0, then V + and V − are linearly independent. Moreover, sec(X, V ) = ε if and
only if V = ±V + or V = ±V − .
Proof. Let u = X × V = (x2 v3 − x3 v2 , x3 v1 − x1 v3 , x1 v2 − x2 v1 ). As X and V are unit
vectors and X ⊥ V , u is a unit normal vector to the plane spanned by X and V . By Lemma

15

2.5, sec(X, V ) = ε if and only if

x2 v3 − x3 v2 = ±m(x1 v3 − x3 v1 ).

In other words, a unit vector V = (v1 , v2 , v3 ) satisfies V ⊥ X and sec(X, V ) = ε if and only
if it satisfies one of the following two linear systems:

x1 v1 + x2 v2 + x3 v3 = 0
(3.1.7)
mx3 v1 − x3 v2 + (x2 − mx1 )v3 = 0
or

x1 v1 + x2 v2 + x3 v3 = 0
(3.1.8)
− mx3 v1 − x3 v2 + (x2 + mx1 )v3 = 0.
To prove the first assertion of (1), use the facts that x3 = 0 and x21 + x22 = 1 to show
V + = ±(x2 , −x1 , 0) where ± is positive if and only if x2 + mx1 > 0. Similarly, V − =
±(x2 , −x1 , 0) where ± is positive if and only if x2 − mx1 > 0.
To prove the second assertion of (1), note that when x3 = 0 and x2 = mx1 , the system
(3.1.7) has a one dimensional solution set spanned by V + . Similarly, when x3 = 0 and
x2 = −mx1 the system (3.1.8) has a one dimensional solution set spanned by V − .
To prove the second assertion of (2), note that system (3.1.7) has a one dimensional
solution since, for instance, the determinant

16

x1

x2

x3

mx3 −x3 x2 − mx1 = x3 = 0.
x3

0

(3.1.9)

−x1

Direct verification shows that the solution set is spanned by V + . The same holds after
replacing system (3.1.7) with (3.1.8), m with −m, and V + with V − .
Finally, as
x1
mx3

x2

x3

−x3 x2 − mx1 = −2mx3 = 0,

−mx3 −x3 x2 + mx1
V + and V − are linearly independent, concluding the proof.
Remark 3.3. Lemma 3.2 implies that a vector X = (x1 , x2 , x3 ) ∈ Up M is unicurved if and
only if x3 = 0 and x2 = ±mx1 , and is generic if and only if x3 = 0. Furthermore, each
generic vector is contained in exactly two curvature ε planes.

3.1.2

Calculations Along Geodesics

Computations in this subsection are mostly done along geodesics. To be more precise, let
γ : [0, δ] → O be a geodesic. We may write γ (t) = (x1 (t), x2 (t), x3 (t)). Quantities such as
V + and V − defined on each tangent space now vary along the geodesic γ(t). To simplify
the notation, we may not indicate that a certain quantity is a function of t. For example,
we write xi instead of xi (t) and m instead of m(γ(t)).
The reader may verify that

K =

mm − (mx2 − x1 )(mx2 − x1 )
K
17

(3.1.10)

and
L =

mm − (−mx2 − x1 )(−mx2 − x1 )
L

(3.1.11)

where K and L are defined in (3.1.3) and (3.1.4).
Lemma 3.4. Let γ : [0, δ] → O be a geodesic such that γ (t) is generic for all t ∈ [0, δ].
Then either V + (t) or V − (t) is a parallel field along γ for all t ∈ [0, δ].
Proof. As M has higher rank, there exists a unit normal parallel field V (t) along γ(t) such
that
sec(γ (t), V (t)) = ε ∀t ∈ [0, δ].
As γ (t) is generic for each t, part (2) of Lemma 3.2 implies that for each τ ∈ [0, δ], V (t) =
±V + (t) or V (t) = ±V − (t). By continuity, the lemma follows.
Lemma 3.5. Let γ : [0, δ] → O be a geodesic whose velocity vectors have components
γ (t) = (x1 (t), x2 (t), x3 (t)). Assume that γ (t) is generic for all t ∈ [0, δ], then V + (t) is
parallel along γ if and only if

{mΓ311 − m2 Γ312 }x21 + {Γ321 − mΓ322 }x22 + {e3 (m) − (1 + m2 )Γ231 }x23
+{Γ311 − mΓ312 + mΓ321 − m2 Γ322 }x1 x2
+{e1 (m) − (1 + m2 )Γ211

− mΓ133

+ m2 Γ233 }x1 x3

+{e2 (m) + (1 + m2 )Γ122 − Γ133 + mΓ233 }x2 x3 = 0
along γ(t).

18

(3.1.12)

Proof. Writing V + (t) = (v1 (t), v2 (t), v3 (t)) along γ(t), we have that
1 + mx1 x2 − x21
K
2
mx2 − x1 x2 − m
v2 =
K
(mx2 − x1 )x3
v3 =
K
v1 =

(3.1.13)

by (3.1.5).
Direct computation using (3.1.3) and (3.1.13) shows that

v1 − mv2 = K.

(3.1.14)

The vector field V + (t) is parallel if and only if it satisfies the parallel field equations:

0=

DV +
=
dt

3

(
k=1

dvk
+
dt

Γkij xi vj )ek ,

(3.1.15)

i,j

where i, j ∈ {1, 2, 3} and Γkij =< ∇ei ej , ek >. In what follows, we prove the lemma by
showing that equations (3.1.15) hold if and only if (3.1.12) holds.
As a first step, substitute (3.1.13) into the undifferentiated terms in (3.1.15) to obtain

mx22 − x1 x2 − m 1
mx22 − x1 x2 − m 1
Γ12 + x2
Γ22
K
K
mx22 − x1 x2 − m 1
x (mx2 − x1 ) 1
+ x3
Γ32 + x3 3
Γ33
K
K
x (mx2 − x1 ) 1
x (mx2 − x1 ) 1
+ x2 3
Γ23 + x1 3
Γ13 = 0,
K
K

v 1 + x1

19

1 + mx1 x2 − x21 2
x (mx2 − x1 ) 2
Γ21 + x2 3
Γ23
K
K
1 + mx1 x2 − x21 2
x (mx2 − x1 ) 2
+ x1
Γ11 + x1 3
Γ13
K
K
1 + mx1 x2 − x21 2
x (mx2 − x1 ) 2
+ x3 3
Γ33 + x3
Γ31 = 0,
K
K

v2 + x2

and
1 + mx1 x2 − x21 3
mx22 − x1 x2 − m 3
Γ11 + x1
Γ12
K
K
mx22 − x1 x2 − m 3
1 + mx1 x2 − x21 3
+ x2
Γ22 + x2
Γ21
K
K
mx22 − x1 x2 − m 3
1 + mx1 x2 − x21 3
Γ31 + x3
Γ32 = 0.
+ x3
K
K

v3 + x1

Rearrange terms to get

1
(mx2 − x1 )(
v1 +
K
v2 +

v3 +

1
(mx2 − x1 )(
K
1
(mx2 − x1 )(
K

3

xi Γ1i2 = 0,

Γ1ij xi xj ) − m
i=1

i,j
3

xi Γ2i1 = 0,

Γ2ij xi xj ) +

(3.1.16)

i=1
3

i,j

xi (Γ3i1 − mΓ3i2 ) = 0.

Γ3ij xi xj ) +
i=1

i,j

As γ is a geodesic, it satisfies the geodesic equations
3

(
k=1

dxk
+
dt

Γkij xi xj )ek = 0.
i,j

Next, use (3.1.17) in (3.1.16) to obtain

20

(3.1.17)

v1 −

3
1
i=1 xi Γi2 )

m(

v2 +

K

(

3
2
i=1 xi Γi1 )

K

−

−

mx2 − x1
x1 = 0,
K

mx2 − x1
x2 = 0,
K

(3.1.18)

(3.1.19)

and
v3 +

3
3
i=1 xi Γi1

−m
K

3
3
i=1 xi Γi2

−

mx2 − x1
x3 = 0.
K

(3.1.20)

Now we claim that equations (3.1.18) and (3.1.19) are equivalent. Indeed, subtract m
times the left hand side of (3.1.19) from the left hand side of (3.1.18), and use equality
(3.1.14) to obtain zero as follows:

mx2 − x1
(x1 − mx2 )
K
mx2 − x1
(x1 − mx2 )
= (v1 − mv2 ) + m v2 −
K
mx2 − x1
(x1 − mx2 )
= K + m v2 −
K
mx2 − x1
1
(mx2 − x1 )
= {mm − (mx2 − x1 )(mx2 − x1 ) } + m v2 +
K
K

(v1 − mv2 ) −

= 0.
Since m = 0, this claim follows.
Next, we will show that (3.1.19) and (3.1.20) are equivalent. To prove this claim, first
differentiate the vi using (3.1.10) and then substitute into (3.1.19) and (3.1.20) to obtain

21

1
(x21 + x22 − 1)m + (mx2 − x1 )(x2 + mx1 )
K3
1
+ (
K

(3.1.21)

3

xi Γ2i1 )

= 0,

i=1

and
1
(x2 + mx1 )x3 m + (1 + m2 )(mx2 − x1 )x3
K3
1
+ (
K

3

(3.1.22)

3

xi Γ3i1

xi Γ3i2 )

−m

i=1

= 0.

i=1

Use (3.1.17) to show that
3

3

3

(3.1.23)

i=1

i=1

i=1

xi Γ3i2 ).

xi Γ3i1 − m

xi Γ2i1 ) − x3 (

mx2 − x1 = −(x2 + mx1 )(

Substitute (3.1.23) into (3.1.21) and (3.1.22) to obtain

1
{x2 m − x23 (1 + m2 )(
K3 3

3

xi Γ2i1 )
i=1

3

xi Γ3i1 − m

+x3 (x2 + mx1 )(

(3.1.24)

3

i=1

xi Γ3i2 )} = 0,
i=1

and
1
{x3 (x2 + mx1 )m − x3 (x2 + mx1 )(1 + m2 )(
3
K
3

+(x2 + mx1 )2 (

3

xi Γ2i1 )
i=1

(3.1.25)

3

xi Γ3i1 − m
i=1

xi Γ3i2 )} = 0.
i=1
x

3
, concluding
Observe that equation (3.1.24) and (3.1.25) differ by the nonzero factor x +mx
2
1

22

the proof of the claim.
Up to now, we have shown that the parallel field equations are equivalent to (3.1.25).
Since K = 0 and γ (t) is generic, (3.1.25) is equivalent to

x3 {(m − (1 + m2 )(x1 Γ211 + x2 Γ221 + x3 Γ231 )
+ (x2 + mx1 )Γ331 − m(x2 + mx1 )Γ332 )}

(3.1.26)

+ (x2 + mx1 )(x1 Γ311 + x2 Γ321 − mx1 Γ312 − mx2 Γ322 ) = 0.
To conclude the proof, note that (3.1.12) follows from (3.1.26), since m = x1 e1 (m) +
x2 e2 (m) + x3 e3 (m).
Remark 3.6. With the same assumption as in Lemma 3.5, an analogous proof replacing m
with −m and K with L shows that V − (t) is parallel if and only if

{−mΓ311 − m2 Γ312 }x21 + {Γ321 + mΓ322 }x22 + {−e3 (m) − (1 + m2 )Γ231 }x23
+{Γ311 + mΓ312 − mΓ321 − m2 Γ322 }x1 x2
+{−e1 (m) − (1 + m2 )Γ211

+ mΓ133

(3.1.27)

+ m2 Γ233 }x1 x3

+{−e2 (m) + (1 + m2 )Γ122 − Γ133 − mΓ233 }x2 x3 = 0
along γ(t).
By Remark 3.3, generic tangent vectors form an open subset in the generic set O. Hence,
for p ∈ O and X = (x1 , x2 , x3 ) ∈ Up M a generic vector, there exists δ > 0 such that the
velocity vectors of the geodesic γ(t) = expp tX are generic for all t ∈ [0, δ]. By Lemmas 3.4
and 3.5, it follows that either (3.1.12) or (3.1.27) hold along γ(t) on [0, δ]. In particular, for
every p ∈ O and generic vector X ∈ Up M , either (3.1.12) or (3.1.27) holds for X at p. This

23

motivates the following definition.
Definition 3.7. Define subsets Bp+ and Bp− of Up M as follows.
(1) X ∈ Bp+ if and only if (3.1.12) holds for X at p.
(2) X ∈ Bp− if and only if (3.1.27) holds for X at p.
Lemma 3.8. For each p ∈ O, Up M ⊂ Bp+ ∪ Bp− .
Proof. Note that the set Bp+ and Bp− are closed subsets of Up M . By Remark 3.3, the generic
vectors are dense in Up M . Hence the remarks preceding Definition 3.7 and the pigeon-hole
principle imply the lemma.

Proposition 3.9. For each p ∈ O, at least one of the following holds:

Γ311 = mΓ312 = mΓ321 = m2 Γ322
e1 (m) = (1 + m2 )Γ211 + mΓ133 − m2 Γ233
e2 (m) =

−(1 + m2 )Γ122

+ Γ133

(3.1.28)

− mΓ233

e3 (m) = (1 + m2 )Γ231 ,
or
Γ311 = −mΓ312 = −mΓ321 = m2 Γ322
e1 (m) = −(1 + m2 )Γ211 + mΓ133 + m2 Γ233
e2 (m) =

(1 + m2 )Γ122

− Γ133

e3 (m) = −(1 + m2 )Γ231 .

24

− mΓ233

(3.1.29)

As Bp+ and Bp− are closed subsets of Up M , Lemma 3.8 implies that at least one of them
contains an interior. Hence to prove the Proposition 3.9, it suffices to prove the following
lemma.
Lemma 3.10. Let p ∈ O. The equation (3.1.28) (respectively (3.1.29)) holds at p if Bp+
(respectively Bp− ) contains an interior.
Proof. We prove the case when Bp+ contains an interior. The proof for the case when Bp−
contains an interior is analogous. If Bp+ contains an open set in Up M , then the degree
two homogeneous polynomial in variables x1 , x2 , x3 defined by the left hand side of (3.1.12)
vanishes identically on an open subset of Tp M . Hence its coefficients all vanish. Consider
the coefficients for x1 x3 , x2 x3 and x23 , we obtain the derivative equations in (3.1.28). As
m = 0, by considering the coefficients for x21 and x22 , we have that

Γ311 = mΓ312

(3.1.30)

Γ321 = mΓ322 .

(3.1.31)

and

Now, (2.1.2), (3.1.30), and (3.1.31) imply that

Γ311 = mΓ312 = mΓ321 = m2 Γ322 ,

as desired.

25

(3.1.32)

3.2

Local Structures on Extremal Sets

This section derives relations between Christoffel symbols at interior points to the extremal
set E. Let p ∈ E and let {e1 , e2 , e3 } be a Ricci diagonalizing orthonormal frame at p that
satisfies (2.1.1).
Recall that from Definition 2.4, we had a disjoint partition E = E+ ∪ E− . We first study
the case when p ∈ E− . The following lemma will show that a vector X = (x1 , x2 , x3 ) is
unicurved if x1 = ±1.
Lemma 3.11. Let p ∈ E− and let X = (x1 , x2 , x3 ) ∈ Up M with x1 = ±1. If V ∈ Up M is
perpendicular to X, then sec(X, V ) = ε if and only if

V =±

1 − x21

−x1 x2

,

1 − x21

,

1 − x21

−x1 x3

.

1 − x21

Proof. Write V = (v1 , v2 , v3 ). We argue like Lemma 3.2. The unit vector

u = X × V = (x2 v3 − x3 v2 , x3 v1 − x1 v3 , x1 v2 − x2 v1 )

is normal to the plane spanned by X and V .
By Lemma 2.5, sec(X, V ) = ε if and only if

x2 v3 − x3 v2 = 0.

Since V ⊥ X, sec(X, V ) = ε if and only if the following system of equations hold:

(v1 , v2 , v3 ) · (0, −x3 , x2 ) = 0

26

(3.2.1)

and
(v1 , v2 , v3 ) · (x1 , x2 , x3 ) = 0,
or equivalently, if and only if
(x1 , x2 , x3 ) × (0, −x3 , x2 )
|(x1 , x2 , x3 ) × (0, −x3 , x2 )|
1 − x21
−x1 x2
−x1 x3
= ±(
,
,
).
2
2
2
1 − x1
1 − x1
1 − x1

V =±

Remark 3.12. Lemma 3.11 and Corollary 2.8 together show that each vector tangent to
E− is either isocurved or univurved. A vector X = (x1 , x2 , x3 ) is unicurved if and only if
x1 = ±1, and is isocurved if and only if x1 = ±1.
Notice that the V in (3.2.1) can be obtained by taking m = 0 in (3.1.5). Likewise, setting
m = 0 in the statement and proof of Lemma 3.5 yields the following:
Lemma 3.13. Let γ : [0, δ] → E− be a geodesic whose velocity vectors have components
γ (t) = (x1 (t), x2 (t), x3 (t)). If γ (t) is unicurved for all t ∈ [0, δ], then

V (t) = (

1 − x21

,

−x1 x2

1 − x21

1 − x21

,

−x1 x3

)

1 − x21

is parallel along γ if and only if

x3 {−x1 Γ211 − x2 Γ221 − x3 Γ231 + x2 Γ331 } + x2 {x1 Γ311 + x2 Γ321 } = 0.

27

(3.2.2)

Proposition 3.14. The following is true for each interior point of E− :

Γ211 = Γ311 = Γ321 = Γ231 = 0

and
Γ133 = Γ122 .
Proof. Let p be an interior point of E− . Then each geodesic starting from p remains in E−
for a short time. Moreover, by Remark 3.12, the unicurved vectors form an open subset of
unit vectors tangent to M near p. Hence, a geodesic starting from p whose initial velocity is
unicurved, has velocity vectors that remain unicurved for a short time.
Let X = (x1 , x2 , x3 ) be a unicurved vector. We claim that

x3 {−x1 Γ211 + x2 (Γ122 − Γ133 ) − x3 Γ231 } + x2 {x1 Γ311 + x2 Γ321 } = 0

(3.2.3)

at the point p.
To prove the claim, let γ(t) = expp (tX). As M has higher rank, there exists a unit
normal parallel field W (t) along γ(t) with sec(γ (t), W (t)) ≡ ε. By Lemmas 3.11 and 3.13,
W (t) = ±V (t) for a short time and the claim follows.
As the unicurved vectors form an open subset of Up M , the degree 2 homogeneous polynomial in variables x1 , x2 , x3 defined by the left hand side of (3.2.3) vanishes identically on
an open subset of Tp M . Hence its coefficients all equal zero, concluding the proof.
Note that only the assumption that Λ = ε was used to prove Proposition 3.14. Hence,
for p ∈ E+ , the same proposition holds after permuting the indices 1 and 2. We record this
in the following proposition.
28

Proposition 3.15. The following is true for each interior point of E+ :

Γ122 = Γ322 = Γ312 = Γ132 = 0

and
Γ233 = Γ211 .

29

Chapter 4
From Local to Global
Recall that a three-manifold of higher rank M = O ∪ E ∪ I, and hyperbolic or spherical rank
rigidity occurs when O = ∅ = E. The goal of this chapter is to study the global structure of
O and E by the Christoffel symbol relations given in Propositions 3.9, 3.14 and 3.15. These
results are the main ingredients in our proof for the rigidity theorems in Chapter 5. More
specifically, we first assume that O is nonempty in section 4.1, and study global structures
of O through evolution equations of along a special family of geodesics. In section 4.2, we
study the local structure of E when O is empty. In the end of section 4.2, we show that
when ε = −1 or 1, E must be empty provided that O is empty. To complete the proof of
rigidity theorems, we show that O = ∅ in Chapter 5.

4.1

Evolution Equations Along Isocurved Geodesics in
Generic Sets

The global structure of O is described in terms of evolution equations of geometric quantities
along a specific family of isocurved geodesics. We begin this section with a construction of
such a family of isocurved geodesics.

30

4.1.1

Existence of Isocurved Geodesics

Let C be a connected component of O. Without loss of generality, we may assume that there
exists a global framing {e1 , e2 , e3 } over C satisfying (2.1.1) as in Remark 2.10. With respect
to this framing, at each point p ∈ C either (3.1.28) or (3.1.29) holds by Proposition 3.9.
Definition 4.1. Define closed subsets F + and F − of C by

F + = p ∈ C | (3.1.28) holds

and
F − = p ∈ C | (3.1.29) holds .
Note that F + ∪ F − = C.
Remark 4.2. The definition of F + and F − depends on the Ricci diagonalizing orthonormal
frame. The reader may verify that points in F + and points in F − are switched if the
orthonormal frame is switched from {e1 , e2 , e3 } to {e1 , −e2 , e3 }.
Lemma 4.3. After possibly switching the frame {e1 , e2 , e3 } to {e1 , −e2 , e3 }, we may assume
that one of the following holds:
(1) F + = F − = C, or
(2) The set C \ F − contains an open set U in C.
Proof. Note that the sets C \ F − and C \ F + are open in C. If both of these sets are empty,
then (1) holds. Otherwise, by Remark 4.2, after possibly switching the frame {e1 , e2 , e3 } to
{e1 , −e2 , e3 }, (2) holds.

31

Convention 4.4. From now on, we always assume that the Ricci diagonalizing frame is
selected such that either (1) or (2) in Lemma 4.3 holds.
Definition 4.5. Define two vector fields on C by

E+ = √

1
1 + m2

E− = √

e1 − √

1

e1
1 + m2

+√

m
1 + m2
m

1 + m2

e2 ,

e2 .

Remark 4.6. Note that these vector fields are isocurved by Corollary 2.8. They are characterized projectively by having the property that every plane containing one of these vectors
has curvature ε.
Lemma 4.7. On F + , ∇E + E + = 0.
Proof. On C, direct calculations shows
1
1
1
< ∇E + E + , e1 >= √
{e1 ( √
) − me2 ( √
)
1 + m2
1 + m2
1 + m2
m
m2
Γ112 + √
Γ122 },
−√
2
2
1+m
1+m

−m
m
{e1 ( √
) + me2 ( √
)
1 + m2
1 + m2
1 + m2
1
m
+√
Γ211 − √
Γ221 },
2
2
1+m
1+m

< ∇E + E + , e2 >= √

1

and

< ∇E + E + , e3 >=

1
{Γ311 − mΓ321 − mΓ312 + m2 Γ322 }.
2
1+m
32

Direct verification using (3.1.28) shows that each of these components is zero, concluding
the proof.
When F + = F − = C, Lemma 4.7 implies that the integral curves of E + are geodesics.
Moreover, by part (3) of Corollary 2.8, these geodesics are isocurved.
In what follows, we construct a family of isocurved geodesics when there exists an open
set U ⊂ C \ F − . To prove the desired result, we first prove the lemma below.
Lemma 4.8. Let p ∈ C \ F − and let X = (x1 , x2 , x3 ) ∈ Up M be a generic vector. Let V + be
the vector perpendicular to X defined by (3.1.5). Then along γX (t) = expp (tX), the parallel
vector field V (t) determined by V (0) = V + satisfies sec(V (t), γX (t)) ≡ ε.
Proof. As p ∈ C \ F − , Lemma 3.10 implies that Bp− has no interior in Up M . By continuity,
it suffices to prove the lemma for generic vectors X ∈ Up M \ Bp− .
Let X = (x1 , x2 , x3 ) ∈ Up M \ Bp− be a generic vector. As M has higher rank, there exists
a normal parallel field V (t) along γX (t) such that sec(V (t), γX (t)) ≡ ε. Since X is generic,
Lemma 3.4 implies that V (t) = V + (t) or V (t) = V − (t) for a short time. If V (t) = V − (t)
for a short time, then by Remark 3.6, (3.1.27) holds for X at p, contrary to the assumption
that X ∈
/ Bp− .
Proposition 4.9. Let p ∈ C \ F − . Then the geodesic α(t) = expp (tE + ) is isocurved.
Proof. As generic vectors are dense amongst the unit vectors perpendicular to E + , it suffices
to prove that each generic unit vector W ⊥ E + generates a parallel vector field W (t) along
α(t) satisfying sec(α (t), W (t)) ≡ ε.
Given W ∈ Up M as above, consider the great circle in Up M :

c(s) = cos (s)E + + sin (s)W.
33

(4.1.1)

For each small s ≥ 0, let γs (t) = expp (tc(s)), and note that α(t) = γ0 (t) and that W = c (0).
Hence, by continuity, it suffices to prove that c (s) generates a parallel vector field Ws (t)
along γs (t) satisfying sec(γs (t), Ws (t)) ≡ ε.
Note that direct computation after substituting (4.1.1) into (3.1.5) shows that c (s) is
−V + associated to c(s). The proposition now follows from Lemma 4.8.
Corollary 4.10. Let p ∈ C \ F − and let α(t) = expp (tE + ). Let I ⊂ R be the largest
connected interval containing zero such that α(I) ⊂ C. Then on I, α = E + .
Proof. Consider the subset T ⊂ I defined by

T = {t ∈ I | α (t) = E + (α(t))}.

By construction, 0 ∈ T , and T is closed. As I is connected, it remains to prove that T is
open.
For s ∈ T , α (t) is isocurved for each t in a neighborhood of s by Proposition 4.9. The
lemma now follows by the continuity of the velocity vectors α (t) and Remark 4.6.
We summarize the results obtained in this subsection in the following proposition.
Proposition 4.11. There exists an open subset U ⊂ C with the property that each integral
curve of E + with initial point in U is an isocurved geodesic.
Proof. As ∇E + E + = 0 on F + , the proposition is obvious when (1) holds in Lemma 4.3.
When (2) in Lemma 4.3 occurs, take U to be an open subset of C \ F − , and the proposition
follows from Proposition 4.9 and Corollary 4.10.

34

4.1.2

Evolution Equations

By Proposition 4.11, there exists a family of isocurved geodesics in C that are tangent to E + .
In this subsection, let α be one of these geodesics constructed in Proposition 4.11 and let I
be the maximal connected open interval in R containing zero such that α(I) ⊂ C. Clearly,
I = ∅. The goal of this subsection is to derive first order differential equations satisfied by
geometric quantities along α on I in Proposition 4.19 below. The geometric quantities are
described in terms of a new framing over C.
Definition 4.12. Define another orthonormal frame {E1 , E2 , E3 } on C by E1 = E + , E3 =
e3 , and
E2 = √

m
1 + m2

e1 + √

1
1 + m2

e2 .

By Remark 2.10, the orthonormal frame {E1 , E2 , E3 } is not Ricci diagonalizing. In what
follows below, we reserve the Christoffel symbol notation Γkij for < ∇ei ej , ek >.
Lemma 4.13. With respect to {E1 , E2 , E3 }, at least one of R(E1 , E2 , E2 , E3 ) or R(E1 , E3 , E3 , E2 )
is nonzero, and
R(E1 , E2 , E3 , E1 ) = 0.
Proof. As E1 is isocurved, for every a, b such that a2 + b2 = 1,

ε = R(E1 , aE2 + bE3 , aE2 + bE3 , E1 )
= a2 R(E1 , E2 , E2 , E1 ) + b2 R(E1 , E3 , E3 , E1 ) + 2abR(E1 , E2 , E3 , E1 )
= ε + 2abR(E1 , E2 , E3 , E1 ).
The lemma now follows from (2.0.1), since {E1 , E2 , E3 } does not diagonalize Ricci.

35

(4.1.2)

Note that by Proposition 4.11,
∇E1 E1 = 0

(4.1.3)

along α on I. By using (4.1.3), the following Christoffel symbols vanish when the geodesic
α enters F − .
Lemma 4.14. Let t ∈ I. If α(t) ∈ F − , then

Γ322 = Γ312 = Γ321 = Γ311 = 0.

(4.1.4)

Proof. First calculate

< ∇E1 E1 , E3 >=

1
{Γ3 − mΓ321 − mΓ312 + m2 Γ322 }.
1 + m2 11

(4.1.5)

Next, use (3.1.29) to reduce (4.1.5) to

< ∇E1 E1 , E3 >=

1
{4m2 Γ322 }.
1 + m2

(4.1.6)

By (4.1.3) and the fact that m is nonzero, it follows that Γ322 = 0. This, upon substitution
into (3.1.29), shows that
Γ322 = 0 = Γ312 = Γ321 = Γ311 .

Lemma 4.15. Along the geodesic α(t) for all t ∈ I, we have that

∇E1 E2 = ∇E1 E3 = 0

36

(4.1.7)

and
< ∇E2 E1 , E3 >= 0.

(4.1.8)

Proof. We first prove (4.1.7). By (4.1.3) and the fact that {E1 , E2 , E3 } is an orthonormal
basis, it suffices to prove that ∇E1 E3 = 0.
Straightforward computation shows that

∇E1 E3 =

1
{(Γ113 − mΓ123 )e1 + (Γ213 − mΓ223 )e2 }.
2
1+m

(4.1.9)

If α(t) ∈ F + , substitute (3.1.28) into (4.1.9) to obtain ∇E1 E3 = 0. If α(t) ∈ F − , substitute
(4.1.4) into (4.1.9) to obtain ∇E1 E3 = 0.
To prove (4.1.8), first calculate

< ∇E2 E1 , E3 >=

1
{mΓ311 − m2 Γ312 + Γ321 − mΓ322 }.
1 + m2

(4.1.10)

The equation (4.1.8) now follows from (3.1.28) when in α(t) ∈ F + , and from (4.1.4) when
α(t) ∈ F − .

Lemma 4.16. We have the following evolution equations along α(t) for all t ∈ I:

E1 < ∇E2 E1 , E2 >= −ε− < ∇E2 E1 , E2 >2

and
E1 < ∇E3 E1 , E3 >= −ε− < ∇E3 E1 , E3 >2 .
Proof. To prove the first equality, first use the fact that E1 is isocurved and the fact that
37

∇E1 E1 = 0 to show

ε = R(E2 , E1 , E1 , E2 ) =< −∇E1 ∇E2 E1 − ∇[E ,E ] E1 , E2 > .
2 1

(4.1.11)

By (4.1.7),
< ∇E1 ∇E2 E1 , E2 >= E1 < ∇E2 E1 , E2 > .

(4.1.12)

Use (4.1.7) and (4.1.8) to show

< ∇[E ,E ] E1 , E2 >
2 1
= < ∇∇

E ,E
E2 E1 −∇E1 E2 1 2

= < ∇∇

E ,E
E2 E1 1 2

= < ∇<∇

>

>
(4.1.13)

E ,E
E2 E1 ,E2 >E2 +<∇E2 E1 ,E3 >E3 1 2

>

= < ∇E2 E1 , E2 >< ∇E2 E1 , E2 > + < ∇E2 E1 , E3 >< ∇E3 E1 , E2 >
= < ∇E2 E1 , E2 >2 .
The first equality is now obtained by substituting (4.1.12) and (4.1.13) into (4.1.11).
An analogous proof shows the second inequality.

Definition 4.17. Define an operator A : E1⊥ → E1⊥ on C by A = ∇E1 .
As E1 has unit length, A is well-defined. In terms of the orthonormal frame {E2 , E3 } for
E1⊥ ,
tr(A) =< ∇E2 E1 , E2 > + < ∇E3 E1 , E3 > .
In Lemma 4.18, we show that A is given by scalar multiplication along α.
38

Lemma 4.18. Along the geodesic α(t) for all t ∈ I, we have that
(1) < ∇E2 E1 , E3 >=< ∇E3 E1 , E2 >= 0
(2) < ∇E2 E1 , E2 >=< ∇E3 E1 , E3 > .
Proof. To prove (1), it suffices to prove < ∇E3 E1 , E2 >= 0, by Lemma 4.15. To prove that
< ∇E3 E1 , E2 >= 0, first use (4.1.3) and (4.1.7) to show that

R(E1 , E3 , E2 , E1 )
= E1 < ∇E3 E2 , E1 > − < ∇[E ,E ] E2 , E1 >
1 3
= E1 < ∇E3 E2 , E1 > + < ∇<∇

E ,E
E3 E1 ,E2 >E2 +<∇E3 E1 ,E3 >E3 2 1

(4.1.14)
>

= E1 < ∇E3 E2 , E1 > +tr(A) < ∇E3 E2 , E1 >
By (4.1.2) and (4.1.14),

E1 < ∇E3 E2 , E1 >= −tr(A) < ∇E3 E2 , E1 > .

Hence if < ∇E3 E2 , E1 >= 0 at α(0), then < ∇E3 E2 , E1 >= 0 along α by the uniqueness of
solutions to ordinary differential equations.
Direct calculations show that

< ∇E3 E2 , E1 >=

e3 (m)
− Γ231 .
2
1+m

Hence, at points in F + , the last equality in (3.1.28) implies that < ∇E3 E2 , E1 >= 0,
concluding the proof of (1).
To prove (2), recall from Lemma 4.16 that < ∇E3 E1 , E3 > and < ∇E2 E1 , E2 > satisfy
39

the same first order differential equation, hence it suffices to prove (2) at a point along α(t).
Direct calculations show that

m
1
Γ331 − √
Γ332 .
< ∇E3 E1 , E3 >= √
1 + m2
1 + m2
Note that

∇E2 E1 =∇ √ m

1+m2

m

e1 + √ 1

e2
1+m2

√

1
1 + m2

e1 − √

m
1 + m2

e2

1
1
)+ √
e2 ( √
)
1 + m2
1 + m2
1 + m2
1 + m2
m2
m
Γ112 −
Γ122 }e1
−
2
2
1+m
1+m
−m
1
−m
m
e1 ( √
)+ √
e2 ( √
)
+{ √
1 + m2
1 + m2
1 + m2
1 + m2
m
1
2 +
+
Γ
Γ221 }e2 .
11
2
2
1+m
1+m
={ √

e1 ( √

1

Hence
1
1
m
{me1 ( √
) + e2 ( √
)}
2
2
1+m
1+m
1 + m2
1
−m
−m
+
{me1 ( √
) + e2 ( √
)}
2
1+m
1 + m2
1 + m2
m
m2
m
−√
{
Γ112 +
Γ122 }
2
2
2
1+m
1+m 1+m
1
m
1
2 +
+√
{
Γ
Γ2 }
11
2
2 21
2
1
+
m
1
+
m
1+m
1
1
=√
(−me1 (m) − e2 (m))
1 + m2 1 + m2
m
m2
1 +√ m
√
−
Γ
Γ122 }
{
12
2
2
2
1+m
1+m
1+m
1
m
1
+
{√
Γ211 + √
Γ221 }.
2
2
2
1+m
1+m
1+m

< ∇E2 E1 , E2 >=

40

(4.1.15)

At points in F + , equality in 2. holds as can be seen after substituting the second and
third equalities in (3.1.28) into (4.1.15).

We are now ready to prove the evolution equations along the geodesic α.
Proposition 4.19. Along the geodesic α(t) for all t ∈ I,

1
E1 (tr(A)) = −2ε − (tr(A))2
2
3
E1 (R(E1 , E2 , E2 , E3 )) = − tr(A)R(E1 , E2 , E2 , E3 )
2
3
E1 (R(E1 , E3 , E3 , E2 )) = − tr(A)R(E1 , E3 , E3 , E2 ).
2

(4.1.16)

Proof. By Lemma 4.16 and Lemma 4.18,

E1 (tr(A)) = E1 (< ∇E2 E1 , E2 > + < ∇E3 E1 , E3 >)
= −2ε− < ∇E2 E1 , E2 >2 − < ∇E3 E1 , E3 >2
1
= −2ε − (tr(A))2 .
2
The first equality in (4.1.16) is obtained by expanding the second Bianchi identity

(∇E3 R)(E1 , E2 , E2 , E1 ) + (∇E1 R)(E2 , E3 , E2 , E1 ) + (∇E2 R)(E3 , E1 , E2 , E1 ) = 0,

and then making simplifications using the Christoffel symbols relations

∇E1 E1 = ∇E1 E2 = ∇E1 E3 = 0

41

< ∇E2 E1 , E3 >=< ∇E3 E1 , E2 >= 0
< ∇E2 E1 , E2 >=< ∇E3 E1 , E3 >
obtained in Lemma 4.15 and Lemma 4.18.
The second equality in (4.1.16) is similarly obtained starting from the second Bianchi
identity

(∇E3 R)(E1 , E2 , E3 , E1 ) + (∇E1 R)(E2 , E3 , E3 , E1 ) + (∇E2 R)(E3 , E1 , E3 , E1 ) = 0.

4.2

Evolution Equations Along Isocurved Geodesics in
Extremal Sets

In this section, we prove rigidity results for three-manifolds of higher rank with extremal
curvature ε. Recall that a three-manifold M of higher rank has extremal curvature ε if and
only if O = ∅. Throughout the section, we assume that O = ∅ and E = ∅. Let C be a
connected component of E. By Proposition 2.12, on C or a double cover of C (still denoted
by C), there exists a complete isocurved geodesic field E. By the definition of E+ and E− ,
C ⊂ E− or C ⊂ E+ .
As the rank assumption passes to covers, we may apply Propositions 3.14 and 3.15 to
derive evolution equations along E geodesics on C. The evolution equations are then applied
to prove rigidity results for three-manifolds of higher rank with extremal curvature. These

42

results will be used in the proof of the main theorems in the last section.
Definition 4.20. Let E be the isocurved geodesic field on C. Define A˜ : E ⊥ → E ⊥ by
A˜ = ∇E.
The trace of A˜ is clearly well-defined and smooth on C.
˜ along the E
Proposition 4.21. On C, we have the following evolution equation for tr(A)
geodesics:
˜ = −2ε − 1 (trA)
˜ 2.
E(trA)
2
Proof. We prove the case when C ⊂ E− . The proof for the case when C ⊂ E+ is similar. Let
{e1 , e2 , e3 } be a Ricci diagonalizing orthonormal frame on E− satisfying (2.1.1). To prove
the proposition, first use Proposition 3.14 to obtain

R(e1 , e3 , e3 , e1 ) = e1 (Γ133 ) − (Γ133 )2 .

By Corollary 2.8, e1 is isocurved, so that R(e1 , e3 , e3 , e1 ) = ε. Therefore,

e1 (Γ331 ) = −ε − (Γ331 )2 .

(4.2.1)

Note that on E− , we may assume that E = e1 . By Proposition 3.14, we have that on E− ,

Γ133 = Γ122 .

43

(4.2.2)

Hence by (4.2.1) and (4.2.2),

˜ = e1 (trA)
˜ = e1 (Γ3 + Γ2 )
E(trA)
21
31
1
˜2
= −2ε − (trA)
2

The following proposition is a special case for the characterization of three-manifolds of
higher spherical rank.
Proposition 4.22. Let M be a complete Riemannian three-manifold of higher spherical rank
and extremal sectional curvature 1, then M has constant sectional curvatures.
Proof. Let M be a three manifold of higher spherical rank and extremal curvature 1. Then
M = E ∪ I. To prove the rigidity result, it suffices to show that E is empty.
Suppose that E is nonempty. Let α : (−∞, ∞) → C be a complete isocurved geodesic
obtained by integrating the complete geodesic field E on C. By Proposition 4.21,

d
˜ = −2 − 1 (trA)
˜2
(trA)
dt
2

(4.2.3)

along α(t). The solution to the differential equation (4.2.3) has a finite time singularity.
This contradicts with the fact that trA˜ is a smooth function along α(t). Hence E is empty,
and M has constant sectional curvature 1.
The evolution equations also lead to a different proof of Theorem 2.13.
Proof. Let M be a complete Riemannian three-manifold of higher hyperbolic rank with
extremal curvature and finite volume.

44

Suppose that E is nonempty. Let C be a connected component of E. Then C and its double
cover have finite volume. By Proposition 4.21, we have the following evolution equation along
each isocurved E geodesic:
d
˜ = 2 − 1 (trA)
˜ 2.
(trA)
dt
2

(4.2.4)

The solution of the differential equation (4.2.4) has a finite forward time singularity when
˜
˜
˜
tr(A)(0)
< −2, and it has a finite backward time singularity when tr(A)(0)
> 2. As tr(A)
˜
is a smooth function, tr(A)(0)
∈ [−2, 2] along each isocurved E geodesic. Hence there exists
˜
˜
an open set U such that either −2 < tr(A)(q)
≤ 2 for all q ∈ U or tr(A)(q)
= −2 for all
q ∈ U.
˜
Let φt denote the flow generated by the complete geodesic field E. When −2 < tr(A)(q)
≤
˜ → 2 on φt (U ) as t → ∞. As
2 for all q ∈ U , (4.2.4) implies that div(E) = tr(A)

div(E) =

d
dvol,
dt φt (U )

the volume of φt (U ) is arbitrarily large as t → ∞, contradicting the assumption that C has
finite volume.
˜
When tr(A)(q)
= −2 for all q ∈ U . An analogous argument shows that the volume for
φ−t (U ) is arbitrary large as t → ∞, contradicting the assumption that C has finite volume.
Hence E is empty, and M is hyperbolic.

45

Chapter 5
Rigidity Theorems
We prove the main theorems in this chapter. In the proof of each theorem, we first exclude
the possibility that O is nonempty. Then we use Proposition 4.22 or Theorem 2.13 to obtain
the desired rigidity result. The proof of Theorem 2 is in section 5.1 and the proof of Theorem
1 is in 5.2.
Let C be a connected component of O as in Remark 2.10.
Definition 5.1. On the connected component C ⊂ O, define functions R2 and R3 by

R2 = R(E1 , E2 , E2 , E3 )

and
R3 = R(E1 , E3 , E3 , E2 ).
Along a geodesic α(t) constructed in Proposition 4.11, we may solve the differential
equations (4.1.16) to obtain
3 t
|Ri (t)| = |Ri (0)| exp −
tr(A)(s)ds
2 0

(5.0.1)

for i = 2, 3.
Recall from Lemma 4.13 that either R2 = 0 or R3 = 0 at each point in C. The following

46

lemma is a useful observation for the proof of both theorems.
Lemma 5.2. Let p ∈ C and let α : (k, l) → C be a maximal integral curve of E1 with
α(0) = p. Furthermore, assume that α(t) is a geodesic.
(1) If l is finite, then limt→l− Ri (α(t)) = 0, for i = 2, 3.
(2) If k is finite, then limt→k+ Ri (α(t)) = 0, for i = 2, 3.
Proof. We prove (1). The proof of (2) is analogous. Suppose that l < ∞, then α(l) ∈ E ∪ I.
As M is complete, we may extend α : (k, l) → C to a complete geodesic α : R → M .
Let {E1 (t), E2 (t), E3 (t)} be the orthonormal framing determined by parallel translating the
orthonormal frame {E1 , E2 , E3 } at α(0) along α(t). By (4.1.7), {E1 (t), E2 (t), E3 (t)} agrees
with the orthonormal framing {E1 , E2 , E3 } of C as long as α(t) ∈ C.
If α(l) ∈ I, then the formula for a curvature tensor at an isotropic point implies that

R(E1 (l), E2 (l), E2 (l), E3 (l)) = 0 = R(E1 (l), E3 (l), E3 (l), E2 (l)).

The lemma now follows from continuity of parallel translation and the curvature tensor.
Next, consider the case when α(l) ∈ E. Then exactly one of λ(t) or Λ(t) converges to
ε as t → l− . If Λ(t) → ε, then m → ∞. The formulae for E1 , E2 and E3 in C then
imply that the distances between the ordered orthonormal frames {E1 (t), E2 (t), E3 (t)} and
{−e2 (t), e1 (t), e3 (t)} converge to zero as t → l− . Continuity of the curvature tensor and
(2.0.1) imply (1) in this case. Likewise, if λ → ε, then m → 0. In this case, the distances between the ordered orthonormal frames {E1 (t), E2 (t), E3 (t)} and {e1 (t), e2 (t), e3 (t)} converge
to zero as t → l+ , concluding the proof similarly.

47

5.1

Spherical Rank Rigidity

For three-manifolds of higher spherical rank, we have the following theorem (Theorem 2):
Theorem 5.3. A complete Riemannian three-manifold M has higher spherical rank if and
only if M has constant sectional curvatures one.
Proof. By Proposition 4.22, it suffices to show that O is empty. Suppose that O is nonempty.
By Proposition 4.11, there exists a geodesic α such that α(0) ∈ F + and α = E1 on a
connected component C of O. Let (k, l) be the maximal open interval containing zero such
that α((k, l)) ⊂ C.
Set ε = 1 in Proposition 4.19 to obtain

1
d
(tr(A)) = −2 − (tr(A))2 .
dt
2

(5.1.1)

If l < ∞, we obtain a contradiction as follows. By Lemma 4.13, either |R2 (0)| > 0 or
|R3 (0)| > 0. Assume that |R2 (0)| > 0. The argument for the case when |R3 (0)| > 0 is
similar. Note that −tr(A) is well-defined and smooth on (k, l). Furthermore, (5.1.1) implies
that −tr(A) is strictly increasing on (k, l). Hence by (5.0.1),

lim |R2 (t)| = lim |R2 (0)| exp −

t→l−

t→l−

3 t
tr(A)(s)ds
2 0

3
> |R2 (0)| exp − tr(A)(0) · l > 0.
2
This is a contradiction to Lemma 5.2.
In the case when l = ∞, a contradiction is obtained as the solution to (5.1.1) has a finite
forward time singularity.

48

5.2

Hyperbolic Rank Rigidity

We first prove the following property for the isocurved geodesics constructed in Proposition
4.11.
Lemma 5.4. Let α(t) be an isocurved geodesic as constructed in Proposition 4.11 and let
(k, l) be the maximal open interval containing zero such that α((k, l)) ⊂ C.
(1) l = ∞ if tr(A)(0) ≥ −2.
(2) k = −∞ if tr(A)(0) ≤ −2.
Proof. By taking ε = −1 in Proposition 4.19, we have the following evolution equation along
α(t) on (k, l):

d
1
(tr(A)) = (2 − tr(A))(2 + tr(A)).
dt
2

(5.2.1)

Note that by Lemma 4.13, either |R2 (0)| > 0 or |R3 (0)| > 0. Assume that |R2 (0)| > 0.
The argument for the case when |R3 (0)| > 0 is similar.
To prove (1), suppose that l < ∞ when tr(A)(0) ≥ −2. A straightforward analysis on the
stability of equilibrium points for (5.2.1) shows that tr(A)(t) is bounded by some constants
on [0, l) when tr(A)(0) ≥ −2. Hence by (5.0.1), we have

lim |R2 (t)| = lim |R2 (0)| exp −

t→l−

t→l−

3 t
tr(A)(s)ds > 0.
2 0

This is a contradiction to Lemma 5.2, concluding the proof of (1).
To prove (2), suppose that k is finite when tr(A)(0) ≤ −2. In this case, a straightforward
analysis on the stability of equilibrium points for (5.2.1) shows that tr(A)(t) is bounded
49

above by -2 on (k, 0]. By (5.0.1),

lim |R2 (t)| = |R2 (0)| exp −

t→k +

3 k
tr(A)(s)ds > 0,
2 0

(5.2.2)

contradicting Lemma 5.2. Hence k = −∞, concluding the proof of (2).

We now give the proof of Theorem 1, which is restated in the following.
Theorem 5.5. A complete finite volume Riemannian three-manifold M has higher hyperbolic
rank if and only if M is hyperbolic.
Proof. By Theorem 2.13, it suffices to show that O is empty. Suppose that O is nonempty.
Let U be the open subset of C as in Proposition 4.11. For each p ∈ U , let ϕt (p) be the
integral curve of E1 starting from p.
Note that after possibly shrinking the open set U , we may assume that either tr(A)(p) >
−2 for all p ∈ U or tr(A)(p) ≤ −2 for all p ∈ U .
If tr(A)(p) > −2 for all p ∈ U , then Lemma 5.4 implies that ϕt (U ) ⊂ C for all t > 0.
Furthermore, (5.2.1) implies that tr(A) approaches to 2 on ϕt (U ) as t approaches to infinity.
As
tr(A) = div(E1 ) =

d
dvol,
dt ϕt (U )

(5.2.3)

the volume of C is infinite, a contradiction to the assumption that M has finite volume.
Hence O is empty.
Likewise, if tr(A)(p) ≤ −2 for all p ∈ U , Lemma 5.4 implies that ϕ−t (U ) ⊂ C for all

50

t > 0. The equation (5.2.1) again shows that

div(−E1 ) = −div(E1 ) → 2

on ϕ−t (U ) as t approaches to infinity. Hence an analogous argument shows that C has infinite
volume, contradicting to the assumption that M has finite volume. Hence O is empty. This
concludes the proof of the theorem.
We now prove Theorem 5.
Theorem 5.6. A homogeneous three-manifold M of higher hyperbolic rank is hyperbolic.
Proof. By Theorem 2.14, it suffices to rule out the case when M = O. Suppose that M = O.
Since M is homogeneous, R2 and R3 are constant on C. At each point in C, either R2 = 0
or R3 = 0. By (4.1.16), tr(A) = 0 on C, contradicting with the first equation in (4.19).

5.3

Euclidean Rank Rigidity

In this section, we show that O must be empty for finite volume three-manifolds of higher
Euclidean rank. This leads to the proof of Theorem 3.
Theorem 5.7. Let M be a finite volume three-manifold of higher Euclidean rank. Then M
has extremal curvatures, i.e., M = E ∪ I.
To prove Theorem 5.7, we first prove the following lemma.
Lemma 5.8. Let M be a finite volume three-manifold of higher Euclidean rank and let α(t)
be an isocurved geodesic as constructed in Proposition 4.11. Then
(1) α((−∞, ∞)) ⊂ C,
51

(2) tr(A) = 0 along α(t) for all t ∈ (−∞, ∞), and
(3) either < ∇E3 E3 , E2 > or < ∇E2 E2 , E3 > is a non-constant linear function of t along
α(t) for all t ∈ (−∞, ∞).
Proof. By setting ε = 0 in Proposition 4.19, we have that

1
d
(tr(A)) = − (tr(A))2
dt
2

(5.3.1)

along α on the maximal open interval I containing 0 such that α(I) ⊂ C. The solution to
(5.3.1) goes to −∞ in finite forward time when trA(0) < 0, ∞ in finite backward time when
trA(0) > 0. As α(0) is a generic point, either R2 (0) = 0 or R3 (0) = 0. Equation (5.0.1)
and Lemma 5.2 together implies that tr(A) develops singularity on I unless trA(0) = 0.
Therefore, trA(t) = 0 on I. The differential equation (4.1.16) then implies that Ri (t) is
constant along α(t) for t ∈ I and i = 2, 3. Using Lemma 5.2 again, I = (−∞, ∞) and
trA(t) ≡ 0, which implies (1) and (2).
To prove (3), first use Lemmas 4.15 and 4.18 and (2) to calculate

R2 = R(E1 , E2 , E2 , E3 ) = E1 (< ∇E2 E2 , E3 >)

and
R3 = R(E1 , E3 , E3 , E2 ) = E1 (< ∇E3 E3 , E2 >).
As either R2 (t) or R3 (t) is a nonzero constant, (3) follows.
We now give the proof of Theorem 5.7.
Proof. Let M be a finite volume three-manifold of higher Euclidean rank. Suppose that O
52

is nonempty. Let U be the open subset of C as in Proposition 4.11. For each p ∈ U , let ϕt (p)
be the integral curve of E1 starting from p. By (1) and (2) of Lemma 5.8, ϕt gives a volume
measure preserving R action on the open set

G=

ϕs (U ).
s∈R

By assumption, G has finite measure. The Poincare recurrence Theorem then implies that
there exists a small compact set K, a point x ∈ K, and a monotonic sequence of real numbers
{ni } to infinity such that ϕni (x) ∈ K. This contradicts with (3) of Lemma 5.8. Hence O is
empty.
By diagonalizing the Jacobi operator, one can see that when M has extremal curvatures,
a normal parallel field making curvature 0 with the geodesic is actually Jacobi. Therefore,
Theorem 3 is a corollary of Theorem 5.7 and the rigidity theorem of Bettiol and Schmidt
[3, Theorem A]. We would like to remind the reader again that the Euclidean rank in [3] is
defined as the least number of independent normal parallel Jacobi fields.

53

Chapter 6
Examples
We now give constructions of non-homogeneous manifolds of higher hyperbolic rank with
infinite volume. Let Hn = {(xi ) ∈ Rn | xn > 0} be the upper half-plane model for the
2
2
hyperbolic space with the metric hn = x−2
n (dx1 + · · · + dxn ). The examples we construct in

this chapter are warped products of R and the hyperbolic space. Throughout, let

M n = R × Hn−1 = {(t, (xi )) | t ∈ R, (xi ) ∈ Hn−1 }.

The main goal of the chapter is to prove a more detailed version of Theorem 4.
Theorem 6.1. For each (a, b) ∈ R × [−1, 1], M n admits a complete Riemannian metric
hn,a,b satisfying the following conditions.
(1) For fixed n ≥ 2, the metrics hn,a,b depend smoothly on (a, b).
(2) There is a surjective homomorphism from Isom(Hn−1 , hn−1 ) → Isom(M n , hn,a,b ).
∂ exponentiates to a totally(3) Each tangent 2-plane to M n containing the tangent vector ∂t

geodesic surface isometric to the hyperbolic plane.
∂ is
(4) The sectional curvature of a 2-plane σ at p = (t, (xi )) making angle θ ∈ [0, π2 ] with ∂t

given by
sec(σ) = −1 +

4(1 − b2 − e2a )
sin2 (θ).
t
−t
2
((1 + b)e + (1 − b)e )
54

(5) For n ≥ 3, (M n , hn,a,b ) Riemannian covers a finite volume manifold if and only if
1 = b2 + e2a , or equivalently, if and only if (M n , hn,a,b ) is isometric to (Hn , hn ).

6.1

Warping hyperbolic 1-space over the Euclidean line.

In this section, we describe a way to write hyperbolic space as a warped product over R. Let
t denote the Euclidean coordinate on R and r > 0 the Euclidean coordinate on H1 . Consider
the foliation of H2 by the family of (Euclidean) upper-half semicircles with common center
the origin in R2 . Each such semicircle, parameterized appropriately, is an h2 -geodesic: For
each r > 0, the map
Fr : R → H2
defined by Fr (t) = (r tanh(t), r sech(t)) parameterizes the semicircle through (0, r) in the
clockwise fashion as a unit-speed h2 -geodesic with the initial point (0, r).
Define a diffeomorphism F : R × H1 → H2 by

F (t, r) = Fr (t).

In the coordinates (t, r) on R × H1 , the metric F ∗ (h2 ) is given by

F ∗ (h2 ) = dt2 + cosh2 (t)h1 = dt2 + e2 ln(cosh(t)) h1 .

Hence, (H2 , h2 ) is isometric to a warped product of the hyperbolic line over the Euclidean
line.
In fact, the hyperbolic plane H2 is isometric to many different warpings of H1 over R.

55

The warping functions are described in terms of solutions to the second order differential
equation

φ (t) + (φ (t))2 − 1 = 0.

(6.1.1)

For a ∈ R and −1 ≤ b ≤ 1, the solution to (6.1.1) determined by the initial conditions
φ(0) = −a and φ (0) = b is given by

(1 + b)et + (1 − b)e−t
φa,b (t) = ln(
).
2ea

(6.1.2)

Let M 2 = R × H1 and let π : M 2 → R be the first coordinate projection. Ignore the
slight abuse of notation in letting φa,b also denote the function π ∗ (φa,b ) on M 2 .
For a and b as above, consider the warped product Riemannian metric

2φ
h2,a,b = dt2 + e a,b h1

on M 2 . In this notation, h2,0,0 = F ∗ (h2 ).
Lemma 6.2. For each (a, b) ∈ R × [−1, 1], (M 2 , h2,a,b ) is isometric to the hyperbolic plane
(H2 , h2 ).
Proof. The proof only uses the fact that φa,b is a solution to (6.1.1). For this reason, and
to simplify notation, set h = h2,a,b and φ = φa,b in the remainder of this proof. As (R, dt2 )
and (H1 , h1 ) are complete, so too is their warped product (M 2 , h) [13, Lemma 40, pg. 209].
Therefore, it suffices to prove that h has constant curvature −1.
The curvature at each point equals −1 as a consequence of (6.1.1) since by [13, Proposition
42 (4), pg. 210], the sectional curvature at a point is given by −φ − (φ )2 .
56

6.2

Main Construction

The main construction is a higher dimensional generalization of the analogue in the previous
section. However, for dimension greater than two, the metrics we obtain are no longer
hyperbolic.
Fix n ≥ 2 and (a, b) ∈ R × [−1, 1]. Let φa,b be as in (6.1.2). Let hn,a,b denote the warped
product Riemannian metric on M n = R × Hn−1 defined by

2φ
hn,a,b = dt2 + e a,b hn−1 .

The metric hn,a,b is complete by [13, Lemma 40, pg. 209]. We now consider the properties
(1)-(5) stated in Theorem 6.1.

Property (1): Property (1) is immediate since the warping functions φa,b depend smoothly
on (a, b) ∈ R × [−1, 1].

Property (2): Given F ∈ Isom(Hn−1 , hn−1 ), define F¯ : M n → M n by

F¯ (t, (xi )) = (t, F (xi )).

Then F¯ ∈ Isom(M n , hn,a,b ) and the map

Isom(Hn−1 , hn−1 ) → Isom(M n , hn,a,b )

defined by F → F¯ is a monomorphism.

57

Property (3): To find one such totally-geodesic hyperbolic plane, define G ∈ Isom(Hn−1 , hn−1 )
by
G(x1 , . . . , xn−2 , xn−1 ) = (−x1 , . . . , −xn−2 , xn−1 ).
¯ is an isometry, its fixed point set
As G

Σ = {(t, (0, . . . , 0, xn−1 ))} ⊂ M n

is a totally-geodesic surface. The map (t, (0, . . . , 0, xn−1 )) → (t, xn−1 ) defines an isometry
between the induced metric on Σ and (M 2 , h2,a,b ). By Lemma 6.2, Σ is isometric to the
hyperbolic plane. We conclude by showing that given an arbitrary tangent plane σ as in
Property (3) there is an isometry of M n that carries a tangent plane to Σ to σ.
Let π : M n → R denote the first coordinate projection and Ft¯ the fiber above t¯ ∈ R. For
each p ∈ Ft¯, let Vp denote the set of tangent 2-planes to M n at p that contain the vector
∂
∂t (p).

Let
Xt¯ =

Vp
p∈Ft¯

and let σ0 denote the tangent space to Σ at the point (t¯, (0, . . . , 0, 1)) and note that σ0 ∈ Xt¯.
As Isom(Hn−1 , hn−1 ) acts transitively on unit-tangent vectors to Hn−1 , Isom(M n , hn,a,b )
acts transitively on the set Xt¯. Hence, there exists an isometry I¯ of M n that carries σ0 to σ.
As
¯ 0 )) = I(exp(σ
¯
¯
exp(σ) = exp(dI(σ
0 )) = I(Σ),
exp(σ) is a totally geodesic surface isometric to the hyperbolic plane.

58

∂ be orthonormal vectors at p.
Property (4): Let p = (t, (xi )) ∈ M and let v, w, ∂t

Property (3) implies that
R(

∂
∂
, w, w, ) = −1.
∂t
∂t

(6.2.1)

By [13, Proposition 42 (5), pg. 210],

R(v, w, w, v) = −1 +

4(1 − b2 − e2a )
.
((1 + b)et + (1 − b)e−t )2

(6.2.2)

By [13, Proposition 42 (3), pg.210],

R(

∂
, w)v = 0.
∂t

(6.2.3)

Now fix a two dimensional subspace σ ⊂ Tp M . Assume that σ makes angle θ ∈ [0, π2 ]
∂ . Then there exist unit length vectors v and w perpendicular to ∂ such that
with ∂t
∂t

{¯
v = cos(θ)

∂
+ sin(θ)v, w}
∂t

is an orthonormal basis of σ. By (6.2.1)-(6.2.3) and the symmetries of the curvature tensor,

sec(σ) = R(¯
v , w, w, v¯)
∂
∂
, w, w, ) + sin2 (θ)R(v, w, w, v)
∂t
∂t
2
4(1 − b − e2a )
sin2 (θ).
= −1 +
((1 + b)et + (1 − b)e−t )2
= cos2 (θ)R(

Property (5): As (M n , hn,a,b ) is simply connected and complete, it is isometric to (Hn , hn )
if and only if it has constant sectional curvatures −1. By (4), this is equivalent to the
parameters satisfying 1 = b2 + e2a . As (Hn , hn ) Riemannian covers finite volume manifolds,
59

it remains to prove that when n ≥ 3 and 1 = b2 + e2a , (M n , hn,a,b ) does not Riemannian
cover a finite volume manifold.
Assume that 1 = b2 + e2a . As isometries preserve sectional curvatures, Property (4)
implies that each isometry F ∈ Isom(M n , hn,a,b ) splits as F ((t, xi )) = (F1 (t), F2 (xi )) for
some F1 ∈ Isom(R, dt2 ) and F2 ∈ Isom(Hn−1 , hn−1 ). If b = ±1, then F1 is the identity. If
√
b = ±1, then F1 is either the identity or reflection about the critical point t0 = ln( 1 − b) −
√
ln( 1 + b) of ((1 + b)et + (1 − b)e−t )2 .
Let (N, g) be a Riemannian manifold with universal covering (M n , hn,a,b ). Its fundamental group π1 (N ) is identified with a subgroup Γ of Isom(M, hn,a,b ) that acts properly
discontinuously and fixed point freely on M n . The association γ → γ1 described above
defines a homomorphism Φ : Γ → Isom(R, dt2 ) whose kernel has index at most two in Γ.
¯ , g¯) be the Riemannian covering of (N, g) associated to the subgroup ker Φ of Γ.
Let (N
As each isometry in ker Φ acts trivially on the first factor of M n = R × Hn−1 , the metric
φ
g¯ is also a warped product metric with the same warping function e n,a,b . By (4), g¯ has
φ
negative sectional curvatures. By [4, Lemma 7.6], e n,a,b is a non-constant convex function

¯ , g¯). By [4, Proposition 2.2], (N
¯ , g¯) has infinite volume. Therefore, (N, g) has infinite
on (N
volume, as required.

60

Chapter 7
Concluding Remarks
Our ultimate goal is to generalize Theorem 1 and 2 into higher dimensions.
Conjecture 7.1. A complete finite volume Riemannian manifold has higher hyperbolic rank
if and only it is locally symmetric. A complete Riemannian manifold has higher spherical
rank if and only if it is locally symmetric.
The full conjecture seems to be hard. Without additional assumptions on the curvature
tensors, it is unlikely that our approach for the three dimensional case can be applied to the
higher dimensional cases. Here in this chapter, we mention several possible directions for
future work.

7.1

Constructing Examples

In Chapter 6, we perturbed umbilic foliations in the real hyperbolic space to obtain a one
parameter family of non-homogeneous manifolds of higher rank. It is natural to ask if one
can generalize this construction to other noncompact rank one symmetric spaces.
Problem 7.2. Is it possible to perturb complex, quaternionic, or octonionic hyperbolic metrics to obtain a one parameter family of non-symmetric metrics of higher hyperbolic rank?

61

7.2

Curvature Assumption

Fixing ε = −1, 0, or 1, it is less known if geometric rank characterizes symmetric spaces
when sec ≥ ε. Spatzier and Strake showed that manifolds of higher rank are not rigid when
sec ≥ 0 by constructing many non-symmetric examples in [17]. Partial rigidity results were
given by Schmidt, Shankar, and Spatzier in [14]. To explain their result, we introduce the
following definitions.
Fix ε = −1, 0, or 1. Let M d be a complete d dimensional Riemannian manifold with
sec ≥ ε. The rank (hyperbolic, Euclidean, or spherical, depending on ε = −1, 0, or 1) of
a geodesic γ : R → M is defined as the number of linearly independent normal parallel
fields making curvature ε with the geodesic. As we assume that the sectional curvatures
are bounded below from ε, these notions of rank are well-defined. We say that the rank
(hyperbolic, Euclidean, or spherical) of M is at least k if the rank of every geodesic in M is
at least k.
Schmidt, Shankar, and Spatzier showed that, when d is odd, if sec ≥ 1 and the spherical
rank of M d is at least d − 2, then M d must have constant sectional curvatures. It is natural
to ask if one can prove a similar result in the hyperbolic rank setting.
Conjecture 7.3. Let M d be a finite volume, complete, d dimensional Riemannian manifold
with sec ≥ −1. Then the hyperbolic rank of M is at least d − 2 if and only if M is real
hyperbolic.
It should be noted that when sec ≥ −1, the hyperbolic rank of the other negatively
curved d dimensional symmetric spaces are less than d − 2.

62

7.3

Stronger Rank assumption

Let M be a complete Riemannian manifold such that every geodesic is contained in a closed,
immersed flat of dimension at least two. Molina and Olmos proved that M is locally symmetric if it is locally irreducible at any point in an open nonempty subset. Motivated by
their results, we give the following definition.
Definition 7.4. A complete Riemannian manifold is said to have higher integrable hyperbolic (respectively spherical) rank if every geodesic in the manifold is contained in an
immersed totally geodesic submanifold of dimension greater than 2 and constant sectional
curvature −1 (respectively 1).
It is natural to ask if the results given by Molina and Olmos can be generalized into the
hyperbolic and spherical setting.
Problem 7.5. Classify manifolds of higher integrable rank.
Here is one small step towards integral spherical rank rigidity.
Theorem 7.6. Let M n be a complete n dimensional Riemannian manifold with n ≥ 3
and let k be an integer satisfying 2k > n. Then M n is isometric to the sphere of constant
sectional curvatures 1 if and only if every geodesic in M n is contained in an embedded totally
geodesic k dimensional sphere of constant sectional curvatures 1.
Proof. Note that all geodesics in M are simple closed, and prime geodesics have length 2π.
Since the index of each geodesic in M is at least k with k greater than n/2, the Bott-Samelson
Theorem [2, Theorem 7.23] implies that M is a homotopy sphere.
Let p ∈ M and γ a unit speed geodesic starting from p. By assumption, γ is contained
in an embedded totally geodesic k dimensional sphere Sγ of constant sectional curvatures 1.
63

Let A(p) be the antipodal point of p in Sγ and let η be any other unit speed geodesic
starting from p. Then η is also contained in an embedded totally geodesic k dimensional
sphere Sη of constant sectional curvatures 1. As 2k > n, the tangent spaces Tp Sγ and Tp Sη
share a line. Hence η also passes through A(p) at time π. Therefore, all geodesics starting
from p passes through A(p) at time π and inj(M ) = diam(M ) = π. Hence M is a Blaschke
manifold homotopic to a sphere. By the solution to the Blaschke conjecture for spheres [2,
Appendix D], M has constant sectional curvatures 1.

64

BIBLIOGRAPHY

65

BIBLIOGRAPHY

[1] Werner Ballmann. Nonpositively curved manifolds of higher rank. Ann. of Math. (2),
122(3):597–609, 1985.
[2] Arthur L. Besse. Manifolds all of whose geodesics are closed, volume 93 of Ergebnisse
der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas].
Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P.
Bourguignon, L. B´erard-Bergery, M. Berger and J. L. Kazdan.
[3] Renato G. Bettiol and Benjamin Schmidt. Three-manifolds with many flat planes.
Trans. Amer. Math. Soc., To appear.
[4] R. L. Bishop and B. O’Neill. Manifolds of negative curvature. Trans. Amer. Math. Soc.,
145:1–49, 1969.
[5] Keith Burns and Ralf Spatzier. Manifolds of nonpositive curvature and their buildings.
´
Inst. Hautes Etudes
Sci. Publ. Math., (65):35–59, 1987.
[6] Chris Connell, Thang Nguyen, and Ralf Spatzier. Hyperbolic rank rigidity for manifolds
of 41 -pinched negative curvature. arxiv: 1705.02437, 05 2017.
[7] Christopher Connell. A characterization of homogeneous spaces with positive hyperbolic
rank. Geom. Dedicata, 93:205–233, 2002.
[8] David Constantine. 2-frame flow dynamics and hyperbolic rank-rigidity in nonpositive
curvature. J. Mod. Dyn., 2(4):719–740, 2008.
[9] Patrick Eberlein and Jens Heber. A differential geometric characterization of symmetric
´
spaces of higher rank. Inst. Hautes Etudes
Sci. Publ. Math., (71):33–44, 1990.
[10] Ursula Hamenst¨adt. A geometric characterization of negatively curved locally symmetric
spaces. J. Differential Geom., 34(1):193–221, 1991.
[11] Robert Maltz. The nullity spaces of curvature-like tensors. J. Differential Geometry,
7:519–523, 1972.
[12] Bernardo Molina and Carlos Olmos. Manifolds all of whose flats are closed. J. Differential Geom., 45(3):575–592, 1997.
[13] Barrett O’Neill. Semi-Riemannian geometry, volume 103 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.
With applications to relativity.
66

[14] Benjamin Schmidt, Krishnan Shankar, and Ralf Spatzier. Positively curved manifolds
with large spherical rank. Comment. Math. Helv., 91(2):219–251, 2016.
[15] Benjamin Schmidt and Jon Wolfson. Three-manifolds with constant vector curvature.
Indiana Univ. Math. J., 63(6):1757–1783, 2014.
[16] K. Shankar, R. Spatzier, and B. Wilking. Spherical rank rigidity and Blaschke manifolds.
Duke Math. J., 128(1):65–81, 2005.
[17] R. J. Spatzier and M. Strake. Some examples of higher rank manifolds of nonnegative
curvature. Comment. Math. Helv., 65(2):299–317, 1990.
[18] Jordan Watkins. The higher rank rigidity theorem for manifolds with no focal points.
Geom. Dedicata, 164:319–349, 2013.

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