UNIVERSITY OF CALIFORNIA RIVERSIDE

A Jacobi Field Splitting Theorem for Positive

A Dissertation submitted in partial satisfaction of the requirements for the degree of

Doctor of Philosophy

in

Mathematics

by

Dennis Michael Gumaer

June 2013

Dissertation Committee:

Dr. Frederick Wilhelm, Chairperson Dr. Yat Sun Poon Dr. Bun Wong Copyright by Dennis Michael Gumaer 2013 The Dissertation of Dennis Michael Gumaer is approved:

Committee Chairperson

University of California, Riverside Acknowledgments

I would like to thank Dr. Frederick Wilhelm for his constant assistance. His well considered comments and brilliant insights were invaluable throughout my education.

I am grateful for the support I received from my whole family. I appreciate the time and effort everyone put forth to make my education possible.

Most importantly I want to thank my wife Danaca. The support and encourage- ment she provided kept me going each and every day. I owe everything to her presence in my life.

iv For Danaca

v ABSTRACT OF THE DISSERTATION

A Jacobi Field Splitting Theorem for Positive Curvature

by

Dennis Michael Gumaer

Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, June 2013 Dr. Frederick Wilhelm, Chairperson

This dissertation will present two new rigidity theorems for manifolds with sectional cur- vature bounded below. The main new result, stated below, is a new splitting theorem for

Jacobi fields on manifolds with positive .

Theorem 1. Let M be an n-dimensional with sec ≥ 1. For α ∈ [0, π) let γ :[α, π] −→ M be a . Let Λ be an (n − 1)-dimensional family of Jacobi fields on which the Riccati operator S is self adjoint. If

max{eigenvalue S(α)} ≤ cot α then Λ splits orthogonally into

span{J ∈ Λ | J has a zero before time π} ⊕ {J ∈ Λ | where J takes the form sin(t) · E(t)}

with E(t) being a parallel field.

vi Contents

List of Figures viii

1 Introduction 1

2 Background 3 2.1 The Curvature Tensor ...... 5 2.2 Curvature ...... 8 2.3 Jacobi Fields ...... 9

3 Recent Results 12

4 New Results 19

Bibliography 34

vii List of Figures

2.1 Sample of a Variation of a Curve ...... 6 2.2 Examples on a Sphere ...... 8 2.3 Jacobi Field along a Geodesic ...... 10

viii Chapter 1

Introduction

Jacobi fields on a Riemannian manifold describe the spread of infinitesi- mally close to a given geodesic. They are named after Carl Jacobi whose work in variational calculus inspired their formulation. Let M be an n-dimensional manifold and γ a geodesic on that manifold. A Jacobi field J along γ manifests as one of the 2n linearly independent solutions to the second order differential equation

J¨(t) + R(J(t), γ˙ (t))γ ˙ (t) = 0

where R is the curvature tensor.

Jacobi fields are valued for their relationship with curvature and geodesics, the two

foundational ideas in . Curvature bends geodesics while maintaining

their desired properties. Jacobi fields can be used to describe this bending. While the

Jacobi fields lie in the tangent bundle of a manifold, when exponentiated they point in the

direction of nearby geodesics. Their magnitude can also give information about how the

geodesics are distributed on the manifold.

1 In 2006 Burkhard Wilking [4] found conditions which allow Jacobi fields to be split into two classes with specific forms. This breakthrough had some significant applications.

There are many other applications in the area of positive sectional curvature which may benefit from a Jacobi splitting theorem. However, one of the conditions for Wilking’s splitting theorem is that sectional curvature is non-negative.

Wilking’s splitting theorem used a rigidity theorem involving manifolds with sec- tional curvature bounded below by zero. Included in this dissertation are new rigidity theorems for manifolds with sectional curvature bounded below by one, and when bounded below by negative one. Via the standard rescaling, this covers all cases for manifolds with sectional curvature bounded below.

The main new result of this manuscript is a splitting theorem for Jacobi fields on manifolds with positive sectional curvature. This characterization allows one to have the benefits of Jacobi fields without the necessity of using the differential equations which define them.

2 Chapter 2

Background

A Riemannian manifold is a topological manifold with an inner product, or Rie- mannian metric, that varies smoothly on the tangent spaces. Let X and Y be vector fields with base points on M. An inner product gp(Xp,Yp), or < Xp,Yp >, on the tangent space

TpM is defined pointwise as

gp : TpM × TpM → R.

The condition that the inner product varies smoothly requires that the function g is smooth

as p ∈ M varies. Each metric determines many important properties. Different metrics can

be assigned to a manifold, thus it is necessary to define a metric along with the manifold.

It is the dependence on the metric that will be the focus of my ongoing research. Details

on this will follow later.

The following will fill in the notation needed for the rest of this section. The length

of a vector v is defined via the metric in the same way the norm is defined in calculus, i.e.

||v|| = hv, vi1/2. Given two points p, q on manifold M, let γ be a curve connecting the two

3 points with γ(0) = p and γ(a) = q.

Definition 2. The length of the curve γ : [0, a] → M is given by the arclength functional

Z a D E1/2 `(γ) = γ(˙t), γ(˙t) dt. 0

Thus the metric defines length of a curve. In an attempt to generalize a line

from Euclidean space we will find the curve which minimizes this length. The procedure of

finding a curve that minimizes the length functional does not produce a unique result; even

if there is only one path which is the shortest distance between the given points. However,

often a unique result can be found up to parameterization.

Definition 3. The curve γ is said to be parameterized by arclength if it has unit speed, or

||γ˙ || = 1.

The arclength functional is a less than ideal method of finding curves which min-

imize arclength because of the ambiguity of parameterization. The energy functional is

better for most purposes.

Definition 4. The length of the curve γ : [0, a] → M which is parameterized by arclength

is given by the energy functional

Z a D E E(γ) = 1/2 γ(˙t), γ(˙t) dt. 0

We now have enough information to define a geodesic.

Definition 5. A curve γ is called a geodesic if it minimizes the energy functional.

Note that this does not preclude the possibility of multiple geodesics which min-

imize the energy functional. See figure 2.2a on page 8. There is an alternate formulation

for a geodesic which is more geometrically intuitive, yet less constructive.

4 Theorem 6. Let γ be a curve parameterized by arclength with its second derivative identi- cally zero, γ¨(t) = 0 for all t. Then γ is a geodesic.

This matches up with the notion that lines are “straight”. A geodesic is a line which follows the manifold and is unchanged by any outside forces. For the remainder of this manuscript γ will denote a geodesic on the n dimensional manifold M.

Given a geodesic γ(t) defined by γ :[a, b] → M there is a variationγ ˜(s, t):

(−, ) × [a, b] → M which maps a family of curves to the manifold. This is the manner in which the term “nearby” geodesics is made precise. See Figure 2.1. This idea is essential for the later formulation of a Jacobi field. The E in the following formulas is the energy functional described above.

The first variation formula: Let γ :(−ε, ε) × [a, b] → M be a smooth variation,

then Z b  2    (s,b) dE(γs) ∂ γ ∂γ ∂γ ∂γ = − 2 , dt + , . ds a ∂t ∂s ∂t ∂s (s,a)

The second variation formula: Let γ :(−ε, ε) × [a, b] → M be a smooth variation of a geodesic γ(t) = γ(0, t). Then

2 Z b 2 2 Z b     d E(γs) ∂ γ ∂γ ∂γ ∂γ ∂γ 2 = dt − R , , dt ds s=0 a ∂t∂s a ∂s ∂t ∂t ∂s  2  b ∂ γ ∂γ + 2 , . ∂s ∂t a

2.1 The Curvature Tensor

In multivariable calculus, the directional derivative gives the rate of change of a

function in a particular direction. The limit process used compares vectors in a neigh-

5  γ(s, t)

0

-

(a) (−, ) × [0, a] (b) The Variation (c) Curves in the Variation on M

Figure 2.1: Sample of a Variation of a Curve borhood of the vector in question. What is usually ignored is the use of the canonical

n isomorphism between R and its tangent space. The two nearby vectors are in different tangent spaces, which are identified with the base space before being compared. In Rie- mannian geometry no such isomorphism exists. A different process is required.

n The ∇·· is the way the complication is resolved. On R , the covariant derivative matches the directional derivative. The fundamental theorem of

Riemannian geometry gives a complete description of the covariant derivative.

Theorem 7 (Fundamental Theorem of Riemannian Geometry). The assignment X → ∇X on (M, g) is uniquely defined by the following properties:

• Y → ∇Y X is a (1,1)-tensor

i.e. ∇αw+βzX = α∇wX + β∇zX

• X → ∇Y X is a derivation

i.e. ∇Y (X1 + X2) = ∇Y X1 + ∇Y X2;

∇Y fX = (DY f)X + f∇Y X.

6 • Covariant differentiation is torsion free

i.e. [X,Y ] = ∇X Y − ∇Y X.

• Covariant differentiation is metric

i.e. DZ < X, Y >=< ∇Z X, Y > + < X, ∇Z Y >.

The curvature tensor of a Riemannian manifold is the (1, 3) tensor defined by:

R(X,Y )Z = ∇X (∇Y Z) − ∇Y (∇X Z) − ∇[X,Y ]Z.

Generally the extra parentheses are omitted and we have

R(X,Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z.

The curvature tensor is also useful as a (0, 4) tensor

hR(X,Y )Z,W i = R(X,Y,Z,W ).

The (0, 4) curvature tensor has some nice symmetries:

1. R is skew-symmetric in the first two and last two entries.

R(X,Y,Z,W ) = −R(Y,X,Z,W ) = R(Y, X, W, Z)

2. R is symmetric between the first two and last two entries.

R(X,Y,Z,W ) = R(Z, W, X, Y )

3. R satisfies a cyclic permutation property called Bianchi’s first identity.

R(X,Y )Z + R(Z,X)Y + R(Y,Z)X = 0.

7 2.2 Curvature

A geodesic is the generalization of a line on Riemannian manifolds. Unlike Eu- clidean geometry, there can be many ways to connect two points with geodesics. Consider the Riemannian manifold S2 with the standard metric, as shown in figure 2.2a. The “stan-

2 3 2 dard” metric on S is the usual metric on R restricted to S . The metric is defined via

the inner product, just as the dot product determines the norm from calculus. The inner

product is applied to vectors in the tangent space, see 2.2b.

N N

S

(a) Geodesics on a sphere (b) Vectors in a Tangent Plane

Figure 2.2: Examples on a Sphere

While there are many types of curvature, the focus here will be on sectional cur- vature and . Defined pointwise, sectional curvature describes how curved the manifold is using two linearly independent tangent vectors x, y emanating from a point p. The span of these linearly independent vectors is a 2-plane, denoted π. The sectional curvature of that 2-plane is denoted sec(π) or sec(x, y). Using the (0, 4) curvature tensor

R, linearly independent vectors x, y ∈ TpM, and < ·, · > the given inner product, sectional

8 curvature is defined by

hR(y, x)x, yi sec(x, y) := . < x, x >< y, y > − < x, y >2

If x and y are orthonormal vectors, this simplifies to

sec(x, y) := hR(y, x)x, yi .

As an example, Sn with the standard, or round, metric has constant sectional curvature

1. At every point of Sn every 2-plane at that point has sectional curvature 1, or sec ≡ 1.

Similarly, if every 2-plane at every point has sectional curvature greater than constant a we say that sectional curvature is bounded below, or sec ≥ a.

Ricci curvature can be determined for each vector in the tangent space by adding

(n−1) sectional of the independent 2-planes with the given vector being one of the two vectors necessary to create the 2-plane. Ricci curvature gives some global information about the manifold and metric in question. The Ricci curvature of a sphere with the round metric is (n − 1).

2.3 Jacobi Fields

Jacobi fields along one geodesic are used to determine the behavior of nearby geodesics. The term “nearby” defined as it is in a variation of geodesics. Refer back to our example of the sphere. See figure 2.3 where the vectors are pointing in the direction of the nearby dashed geodesic. Note that each vector is not on the sphere, but in the tangent space at the base point. One Jacobi field along a geodesic from the north pole to the south pole has its length defined by sin(t), with t ∈ [0, π]. At the north pole, t = 0 we have sin(0)

9 N

S

Figure 2.3: Jacobi Field along a Geodesic and nearby geodesics coincide. As a function, sin(t) becomes larger until it reaches π/2, its maximum value. On the globe this corresponds to geodesics from the north pole to the south pole getting further away from each other until the equator. Beyond that, they start to get closer together until they reach the south pole and meet. This matches up with sin(t)

getting smaller until t = π where sin(t) is zero again.

In the previous example, the Jacobi field’s length was sin(t). The Jacobi field itself

is sin(t) · E(t) where E(t) is a parallel field, i.e. E˙ ≡ 0. As mentioned earlier, there are

2n linearly independent solutions to the Jacobi field equation. Thus sin(t) · E(t) is not the only Jacobi field along γ.

n For n ≥ 3, let γ be a geodesic on S . Also let E1 and E2 be two orthonormal parallel vector fields along γ. Then J = sin(t) · E1(t) + cos(t) · E2(t) is also a Jacobi field.

This Jacobi field has no zeros, as |J| ≡ 1. This describes an unfortunate fact about Jacobi

fields. A generic Jacobi field does not have the geometric properties which makes them

desirable. The wording of the following definition will bear this out.

Definition 8. Let γ be a geodesic defined at least on [a, b] with γ(a) = p and γ(b) = q. If

10 there is a nonzero Jacobi field J along γ such that |J(a)| = |J(b)| = 0 then p and q are said to be conjugate points.

Knowing where the conjugate points lie using Jacobi fields is not an easy propo- sition. It would involve searching the (n − 1)-dimensional space of orthogonal Jacobi fields with J(0) = 0 for the right Jacobi field. The right Jacobi field being the one which has a zero closest to the previous zero.

One of the more famous applications of Jacobi fields is the Rauch Comparison

Theorem.

n n+k Theorem 9 (Rauch, 1951). Let γ : [0, a] → M and γe : [0, a] → Mf with k ≥ 0 be geodesics with the same velocity. Also let J and Je be Jacobi fields along γ and γe respectively such that

D E 0 0 0 0 0 0 J(0) = Je(0) = 0, J (0), γ (0) = Je (0), γe (0) , J (0) = Je (0) .

Assume that γe has no conjugate points on (0, a]. Also assume that for every t and

every x ∈ T M, x˜ ∈ T M the sectional curvatures satisfy the inequalities γ(t) γe(t) f

0 0 ˜ sec(x, γ (t)) ≤ sec(x,e γe (t)) then J ≤ |J| .

One problem with this comparison theorem is that it only holds until the first

conjugate point. Results without this restriction are desired.

11 Chapter 3

Recent Results

The necessary general background for Riemannian geometry is complete. Now we move into specific background for use the New Results chapter.

By definition, a Jacobi field is defined to be the solution to the second order differential equation J 00(t) + RJ(t), γ0(t)γ0(t) = 0. Let S be the Riccati operator, i.e.

SX = X0 for vector field X. In that light, the differential equation can be rewritten:

−RJ(t), γ0(t)γ0(t) = J 00(t) = (SJ)0(t)

= (S0J)(t) + (SJ 0)(t)

= (S0J)(t) + (S2J)(t)

= (S0 + S2)(t) · J(t).

This results in the Riccati equation S2+S0+R = 0. Let Λ be an (n−1)-dimensional

family of normal Jacobi fields on which the associated Riccati operator S is self-adjoint, i.e.

hSX,Y i = hX,SY i for all X,Y ∈ Λ. This is a reasonable restriction, which can be seen by

considering the function f(t) = hSX,Y i (t) − hX,SY i (t). Its derivative f 0(t) is identically

12 zero, thus f(t) is constant. If there is an X,Y with f(t) = 0 then it is zero for the whole family Λ and therefore S is self-adjoint.

Take Ψ ⊂ Λ where

Ψ := {J ∈ Λ | J(t) = 0 for some t}.

Define

V (t) := {J(t) | J ∈ Ψ} ⊕ {J 0(t) | J ∈ Ψ,J(t) = 0} and consequently define

H(t) := V (t)⊥ ∩ γ˙ (t)⊥,

the orthogonal complement of V (t).

When the Riccati operator is non-singular, which occurs on a dense set, define

At : V (t) → H(t)

by

0 h AtJ(t) = J (t)

for J ∈ Ψ, where the superscript h denotes the component contained in H(t). Then extend

At to A defined on all of [0, a].

Theorem 10 (Wilking, 2006). Let J ∈ Λ \ Ψ and define Y (t) := J h(t). Then Y satisfies

the following Jacobi equation

(∇h)2 Y (t) + R(Y (t), γ˙ (t))γ ˙ (t)h + 3AA∗Y (t) = 0. ∂t2

h Proof. Let Xi be ∇ parallel fields with X1(t0) = J(t0). Since J(t0) is horizontal, we also

have J(t0) = Y (t0). Given the A tensor above, note the following formulas

0v ∗ J (t0) = A (t0)J(t0) (3.1)

13 0 − ∗ Xi(t) = A (t)Xi(t) (3.2)

J(t0) = X1(t0) = Y (t0). (3.3)

h Using these formulas and a generic ∇ parallel field Xk we have a lengthy and detailed calculation. Starting with the left most term in the desired equation and taking the inner product with Xk we have

h0h0 h0h 0 h0h 0 < J ,Xk > (t0) =< J ,Xk > (t0)− < J ,Xk > (t0)

h0h 0 0 =< J ,Xk > (t0) (Since Xk is vertical)

h0 0 =< J ,Xk > (t0) (Both are horizontal)

h 00 h 0 0 =< J ,Xk > (t0)− < J ,Xk > (t0)

h 00 =< J ,Xk > (t0)

00 =< J, Xk > (t0)

0 0 0 0 =< J ,Xk > (t0)+ < J, Xk > (t0)

00 0 0 0 0 00 =< J ,Xk > (t0)+ < J ,Xk > (t0)+ < J ,Xk > (t0)+ < J, Xk > (t0)

00 0 0 00 =< J ,Xk > (t0) + 2 < J ,Xk > (t0)+ < J, Xk > (t0)

− 0 0 0v 0h − ∗ 00 =< R J, γ γ ,Xk > (t0) + 2 < J + J , A Xk > (t0)+ < X1,Xk > (t0)

(Using the above formulas and the definition of Jacobi fields)

− 0 0 0v − ∗ 0 0 =< R J, γ γ ,Xk > (t0) + 2 < J , A Xk > (t0)+ < X1,Xk > (t0)

0 0 − < X1,Xk > (t0)

− 0 0 ∗ − ∗ =< R J, γ γ ,Xk > (t0) + 2 < A J, A Xk > (t0)

− ∗ − ∗ − < A X1, A Xk > (t0)

14 − 0 0 ∗ ∗ =< R J, γ γ ,Xk > (t0) − 3 < A J, A Xk > (t0)

− 0 0 − ∗ =< R J, γ γ ,Xk > (t0)+ < 3AA J, Xk > (t0).

Finally we get

0h0h − 0 0 − ∗ < J ,Xk > (t0) =< R J, γ γ ,Xk > (t0)+ < 3AA J, Xk > (t0).

So we have (∇h)2 Y (t) + R(Y (t), γ0(t))γ0(t)h + 3AA∗Y (t) = 0. ∂t2

Let Sˆ be the Riccati operator S restricted to H(t). Then Sˆ is itself a Riccati oper-

ator, which will be proven later, and J h0 = SJˆ . The above horizontal Jacobi equation can be decomposed similarly to the original Jacobi equation. The horizontal Riccati equation

3.5 follows

(∇h)2 Y (t) + R(Y (t), γ0(t))γ0(t)h + 3AA∗Y (t) = 0 ∂t2

or

Y 00 + RY + 3AA∗Y = 0 (3.4)

becomes

Sˆ0 + Sˆ2 + Rh + 3AA∗ = 0. (3.5)

The following lemma is instrumental in the proofs of Theorems 12, 14 and 18.

2 tr(S) Ric+|S0| Lemma 11. Let s := (n−1) and r := n−1 where S0 is the traceless part of S. Then

s2 + s0 + r = 0.

15 Proof. Recall the Riccati equation S0 + S2 + R = 0. As in [2], decompose S into

tr(S) S = · Id + S (n − 1) 0

where S0 is the traceless part of S. Define both

tr(S) Ric + |S |2 s := and r := 0 . (3.6) (n − 1) n − 1

Substituting,

 tr(S) 0 S0 = · Id + S (n − 1) 0

0 = (s · Id + S0)

and

 tr(S) 2 S2 = · Id + S (n − 1) 0

2 = (s · Id + S0) .

Thus S0 + S2 + R = 0 becomes

0 2 (s · Id + S0) + (s · Id + S0) + R = 0

Taking the trace of both sides leaves

 0 2  tr (s · Id + S0) + (s · Id + S0) + R = 0

0 2 tr (s · Id + S0) + tr (s · Id + S0) + tr(R) = 0

0 0 2 s · trId + trS0 + tr (s · Id + S0) + Ric = 0

0 2 s (n − 1) + tr (s · Id + S0) + Ric = 0. (3.7)

16 The middle term needs further simplification. Place the inner product on the space of self- adjoint operators. Note that hS0, Idi = tr(S0 · Id) = tr(S0) = 0 due to S0 being traceless.

The middle term can be rewritten

2 tr (s · Id + S0) = tr ((s · Id + S0) ◦ (s · Id + S0))

= hs · Id + S0, s · Id + S0i

= hs · Id, s · Idi + 2 hS0, s · Idi + hS0,S0i

2 2 = s hId, Idi + |S0|

2 2 = s · trId + |S0|

2 2 = s (n − 1) + |S0| .

Substituting this into the final equation 3.7

0 2 s (n − 1) + tr (s · Id + S0) + Ric = 0

0 2 2 s (n − 1) + s (n − 1) + |S0| + Ric = 0.

Finally, divide both sides by (n − 1) and substitute in r

|S |2 + Ric s0 + s2 + 0 = 0 n − 1

s0 + s2 + r = 0. (3.8)

The following two theorems are stated for reference.

Theorem 12. Let Λ be an (n − 1)-dimensional space of Jacobi fields orthogonal to the geodesic γ : R → M on which the Riccati operator S is self-adjoint. Suppose that {J(t) | J ∈

17 ⊥ Λ} spans γ˙ (t) for all t ∈ R. If Ric(γ ˙ ) ≥ 0 then S ≡ 0 and Ric(γ ˙ ) ≡ 0. In particular, Λ consists of parallel Jacobi fields.

Theorem 13 (Wilking[4]). Let M be an n-dimensional manifold with nonnegative sectional curvature. Consider an (n − 1)-dimensional family Λ of Jacobi fields orthogonal to the geodesic γ on which the associated Riccati operator S is self-adjoint. Then

Λ = spanR{J ∈ Λ | J(t) = 0 for some t} ⊕ {J ∈ Λ | J is parallel}.

18 Chapter 4

New Results

We begin with a rigidity theorem when Ricci curvature is bounded below by −(n−

1).

Theorem 14. For α ∈ [0, ∞) , let Λ be an (n − 1)-dimensional family of Jacobi fields

orthogonal to the geodesic γ :[α, ∞) → M on which the Riccati operator S is self-adjoint.

⊥ Let L = {J(t) | J ∈ Λ} and assume that L spans γ˙ (t) for all t ∈ (α, t0). If

Ric ≥ − (n − 1) ,

max{eigenvalue S (α)} ≤ coth α, and

min{eigenvalue S (t0)} ≥ coth t0 for some t0 ∈ [α, ∞)

0 then S ≡ coth(t) · Id on [α, t0] and consequently sec (γ (t) , ·) ≡ −1 on [α, t0].

A few preliminary propositions to aid in the proof of Theorem 14 are in order.

Proposition 15. Assume the hypotheses of Theorem 14. If, using the notation from 3.6,

0 s(t) ≡ coth(t) on [α, t0] then S(t) ≡ coth(t) · Id on [α, t0]. Consequently sec (γ (t) , ·) ≡ −1.

19 Proof. Since s(t) ≡ coth(t), for t ∈ [α, ∞), s0 + s2 + r = 0 would be

0 = − csch2(t) + coth2(t) + r

= 1 + r

and r ≡ −1. Consequently

Ric + |S |2 r = −1 = 0 n − 1 (1 − n) + |S |2 ≥ 0 n − 1 |S |2 = −1 + 0 n − 1

tr(S) and |S0| ≡ 0. So S = (n−1) · Id + 0 and S ≡ coth(t) · Id. This S in the Riccati equation

determines that the curvature tensor R ≡ −Id, hence the desired sec (γ0 (t) , ·) ≡ −1.

Proposition 16. Assume the hypotheses of Theorem 14. Let f be the solution of the differential equation

0 2 f + f − 1 = 0; f(t0) = s(t0).

The model function f has an asymptote, denoted d, which lies in the interval [0, t0). Addi- tionally, each of the following holds.

• The asymptote d = 0 if and only if s(t0) = coth(t0) if and only if f(t) ≡ coth(t).

• When d ∈ (0, t0), f(t) > coth(t) on (d, t0).

Proof. Let

s(t0) + 1 c = 2t , e 0 (s(t0) − 1) then f satisfies the following equalities:

20 1) This first characterization, since c is constant, is useful for verifying that f is a solution

to the differential equation. ce2t + 1 f(t) = (ce2t − 1)

2) This second equality it can more easily be seen that f satisfies the initial condition.

 s(t0)+1  2t  2t e + 1 e 0 (s(t0)−1) f(t) =    s(t0)+1 2t 2t e − 1 e 0 (s(t0)−1)    s(t0)+1 e2t + (s(t ) − 1) e2t0 0 =    s(t0)+1 e2t − (s(t ) − 1) e2t0 0 (s(t ) + 1) e2(t−t0) + s(t ) − 1 = 0 0 2(t−t ) (s(t0) + 1) e 0 − s(t0) + 1

3) From this last equality, derived from the first with c = elog c, it is easy to see that the

graph of f is a horizontal translation of the graph of coth(t).

 2(t+ 1 log c)  e 2 + 1 f(t) =  2(t+ 1 log c)  e 2 − 1

By hypotheses f(t0) = s(t0) ≥ coth(t0). From this information, a range for the asymptote, denoted d, of f can be identified. The proof that d ∈ [0, t0) is implied by the

following.

The asymptote d = 0 if and only if f(t0) = s(t0) = coth(t0) since f and coth(t)

would solve the same differential equation with the same initial condition. This also only

occurs if and only if f(t) ≡ coth(t).

Recall that coth(t) is a decreasing function on its domain. If f(t0) = s(t0) > coth(t0) then the graph of f must be a horizontal translation of the graph of coth(t) to the right. The endpoint t0 is not possibility for d since s(t0) is defined everywhere and

21 s(t0) = f(t0). Similarly, d cannot be larger than t0 since that would imply that f(t0) < −1.

Thus the asymptote for f must be in (0, t0).

Proposition 17. Assume the hypotheses of Theorem 14 and define f as in Proposition

16. Then on the interval [α, t0), we have the inequality s ≥ f. Furthermore, if f > 0 then

s0 ≤ f 0.

Proof. Set y = f − s. Then

y0 = f 0 − s0

= −f 2 + s2 + r + 1

= −(f − s)(f + s) + r + 1

so y is a solution to

0 y = −(f + s)y + r + 1, y(t0) = 0.

Let x be a non-trivial solution to x0 = −(1/2)(f + s)x and u a function satisfying

0 2 2 u = 2(r + 1)/x with u(t0) = 0. Combining, y = (1/2)ux . This can be verified to be the

solution to the above differential equation by taking the derivative and substituting u0, x, and x0.

0 Note that u ≥ 0 and u(t0) = 0. On the interval [α, t0) must be u ≤ 0 which implies that y ≤ 0. Thus s ≥ f.

Next assume f > 0. Recall that s0 + s2 + r = 0, f 0 + f 2 − 1 = 0, and r ≥ −1. This

implies that −s0 = s2 + r ≥ s2 − 1. Then

s2 ≥ f 2

s2 − 1 ≥ f 2 − 1

22 s2 + r ≥ f 2 − 1

−s0 ≥ −f 0

s0 ≤ f 0

We are now ready to prove Theorem 14.

Proof of Theorem 14. Assume, for contradiction, that S 6≡ coth(t) · Id. It follows from

Proposition 15 that s(t) 6≡ coth(t). Consequently, assume there is a t ∈ [α, t0) such that s(t) 6= coth(t).

Given the hypothesis min{eigenvalue S (t0)} ≥ coth t0 for some t0 ∈ [α, ∞), there

are two cases: s(t0) = coth(t0); s(t0) > coth(t0).

Assume that s(t0) = coth(t0). By Proposition 17, on [α, t0) s ≥ f. The chain of

inequalities

f(α) ≤ s(α) ≤ coth(α) = f(α)

results in the equality s(α) = coth(α). This combined with s(t0) = coth(t0) gives equality

at both ends of the interval [α, t0]. Then there is at least one interval (t1, t2) ⊆ [α, t0]

where s(t) > f(t) = coth(t) on the interior and s = f on the boundary points. On [α, t0],

f(t) = coth(t) > 1 > 0 so Proposition 17 applies and s0 ≤ f 0. Collecting the information,

0 0 s(t1) = f(t1) and s ≤ f on an interval containing t1. This implies that s ≤ f on some

interval (t1, t1 + ) for a sufficiently small  > 0. Yet on [α, t0) s ≥ f. This contradicts the

assumption that there is a t ∈ [α, t0) such that s(t) 6= coth(t).

Next there is the case where s(t0) > coth(t0). Assume d ∈ [α, t0). On [d, t0),

s ≥ f. As t approaches d from the left, f approaches infinity. Consequently s approaches

infinity. Thus s is not defined for all t ∈ [α, t0]. A contradiction.

23 Still in the case where s(t0) > coth(t0), but now assuming d ∈ (0, α). Combine the Propositions 17, 16 and the hypothesis on the eigenvalue of S(α). This results in

s(α) ≥ f(α) > coth(α) ≥ s(α) a contradiction.

This refutes the original assumption that there is a t ∈ [α, t0] such that s(t) 6= coth(t).

Finally, we have s(t) ≡ coth(t). Therefore, by Proposition 15 S ≡ coth(t) · Id and sec(γ0(t), ·) ≡ −1.

The new results will continue with a rigidity theorem for Ric ≥ (n − 1).

Theorem 18. For α ∈ [0, π) , let Λ be an (n − 1)-dimensional family of Jacobi fields orthogonal to the geodesic γ :[α, π] → M on which the associated Riccati operator S is self-adjoint. Let L ≡ {J(t)|J ∈ Λ} and assume that L spans γ˙ (t)⊥ for t ∈ (α, π). If

Ric ≥ n − 1 and

max{eigenvalue S (α)} ≤ cot α

then S ≡ cot(t) · Id and consequently sec(γ, ˙ ·) ≡ 1.

As in Theorem 14, a few propositions will aide in the proof.

Proposition 19. Assume the hypotheses of Theorem 18. If s(t) ≡ cot(t) then S(t) ≡ cot(t) · Id and therefore sec(γ0(t), ·) ≡ 1.

24 Proof. Since s(t) ≡ cot(t) for t ∈ (α, π), s0 + s2 + r = 0 would be

0 = − csc2(t) + cot2(t) + r

= −1 + r and r ≡ 1. Consequently

Ric + |S |2 r = 1 = 0 n − 1 (n − 1) + |S |2 ≥ 0 n − 1 |S |2 = 1 + 0 n − 1

and |S0| ≡ 0. So S = tr(S)/(n − 1) · Id + 0 and S ≡ cot(t) · Id. Substituting this S into the Riccati equation gives that the curvature tensor R ≡ Id and therefore sec(γ0, ·) ≡ 1.

Proposition 20. Assume the hypotheses of Theorem 18. Let f be the solution to the

differential equation

0 2 f + f + 1 = 0; f(t0) = s(t0)

for a t0 to be specified later. The model function f takes the form

−1 f(t) = tan(t0 − t + tan (s(t0)))

and the graph of f is a horizontal translation of cot(t). If s(t0) < cot(t0) then f has an

asymptote in the interval (t0, π). If s(t0) > cot(t0) then f has an asymptote in the interval

(0, t0).

Proof. The solution is easily verified by inspection.

25 If s(t0) < cot(t0) the initial condition f(t0) = s(t0) implies that the graph of f is

the graph of cot(t) shifted to the left. This shift can move the asymptote for f only as far

as, but not including, t0. To go past t0 would require that f(t0) > cot(t0) contradicting

that s(t0) < cot(t0).

Similarly, if s(t0) > cot(t0) f must be cot(t) shifted to the right. Thus the asymp-

tote can go only as far as, but not including, t0.

Proposition 21. Assume the hypotheses of Theorem 18 and define f as in Proposition 20.

Then on the intervals

(a) [α, t0), we have s ≥ f

(b) (t0, π), we have s ≤ f.

Proof. Set y = f − s on (α, t0). Then

y0 = f 0 − s0

= −f 2 + s2 + r − 1

= −(f − s)(f + s) + r − 1 so y is a solution to

0 y = −(f + s)y + r − 1, y(t0) = 0.

Let x be a non-trivial solution to x0 = −(1/2)(f + s)x and u a function satisfying

0 2 2 u = 2(r − 1)/x with u(t0) = 0. Combining, y = (1/2)ux .

For part (a), on the interval [α, t0), u ≤ 0 which implies that y ≤ 0. Thus s ≥ f.

For part (b), on the interval (t0, π), u ≥ 0 which implies that y ≥ 0. Thus

s ≤ f.

26 We are now ready to prove Theorem 18.

Proof of Theorem 18. Assume, for contradiction, that S 6≡ cot(t) · Id. It follows from

Proposition 19 that s(t) 6≡ cot(t). Let t0 be a time such that s(t0) 6= cot(t0).

Assume for the moment that s(t0) < cot(t0). Then, by Proposition 20, f has an

asymptote which occurs in (t0, π). On that interval, by Proposition 21(b) s ≤ f. Yet as t

approaches d from the left, f approaches −∞. So s cannot be defined for all t ∈ (α, π). A

contradiction.

Since there is no t0 with s(t0) < cot(t0), consider the case where s(t) ≥ cot(t) on

[α, π). We have the assumption that

max{eigenvalue S (α)} ≤ cot α.

Thus s(α) ≥ cot(α) ≥ s(α). Hence s(α) = cot(α).

Next, assume that s(t0) > cot(t0). Thus, by Proposition 20, the graph of f is the

graph of cot(t) shifted to the right. If the asymptote for f lies in (0, α) Proposition 21(a)

states that s ≥ f on [α, t0). Combining, s(α) ≥ f(α) > cot(α) = s(α). A contradiction.

The remaining case is when s(t0) > cot(t0) and the asymptote for f is in (α, t0).

On (t0, π) Proposition 21(b) states that s ≤ f. As t approaches the asymptote for f

from the left, f approaches negative infinity. So s cannot be defined for all t ∈ (α, π). A

contradiction.

Hence we have s(t0) ≡ cot(t0). Therefore, by Proposition 19 S ≡ cot(t) · Id and

sec ≡ 1 as desired.

Through the standard rescaling, Theorem 14 covers all cases when Ricci curva- ture is bounded below by an arbitrary negative number. Similarly, Theorem 18 covers all

27 cases when Ricci Curvature is bounded below by a positive number. These combined with

Theorem 12 cover all possible cases for Ricci curvature bounded below.

Now for the proof of the main result, a Jacobi field splitting theorem for positive curvature.

Theorem 22. Let M be an n-dimensional Riemannian manifold with sec ≥ 1. For α ∈

[0, π) let γ :[α, π] −→ M be a geodesic. Let Λ be an (n − 1)-dimensional family of Jacobi

fields on which the Riccati operator S is self adjoint. If

max{eigenvalue S(α)} ≤ cot α then Λ splits orthogonally into

span{J ∈ Λ | J has a zero before time π} ⊕ {J ∈ Λ | where J takes the form sin(t) · E(t)} with E(t) being a parallel field.

Proof. For ease of notation, denote Z := span{J ∈ Λ | J has a zero before time π}. Con- sider

 0 V (t) = {J (t)| J ∈ Z} ⊕ J (t) J ∈ Z,J (t) = 0 .

Also let

H (t) = V (t)⊥ ∩ γ0 (t)⊥ .

For J ∈ Λ we decompose

J (t) = J (t)V (t) + J (t)H(t) , where J (t)V (t) ∈ V (t) and J (t)H(t) ∈ H (t) .

28 Take a non-trivial J ∈ Λ. Choose t0 ∈ [α, π] with J(t0) 6= 0. Define the operator Sˆ by,

 0H(t)  H(t) H(t)0H(t) Sˆ : H(t) → H(t) by Sˆ(t0)J(t0) := J = J .

As in [2], Sˆ is well defined and self adjoint as demonstrated below. Let L(t) ∈ H(t).

Let J ∈ Λ with J(t1) ∈ H(t) a non-zero vector for t1 ∈ [α, π]. Jacobi field J is in Λ, yet

the definition only discusses the part in H(t). It must be shown that the V (t) part has no

effect on Sˆ. Thus, to prove Sˆ is well defined consider

D V (t)0H(t) E D V (t)0 E J ,L (t1) = J ,L (t1)

d hD V (t) Ei D V (t) 0E = J ,L − J ,L (t1) dt t1

= 0.

This last equality holds because both terms are zero. The first because J V (t) is in V (t) and

V (t) L is in H(t), two spaces orthogonal to each other. The second because J (t1) = 0.

Now for the proof that Sˆ is self adjoint. Let J, L ∈ Λ with non-zero vectors

J(t1),L(t1) ∈ H(t) for t1 ∈ [α, ∞). Then

D E D H(t)0H(t) E SJ,ˆ L (t1) = J ,L (t1)

D H(t)0 E = J ,L (t1)

D H(t)0 V (t)0 E = J + J ,L (t1)

= hSJ, Li (t1)

where S is the Riccati operator on Λ. Working back gives

hSJ, Li (t1) = hJ, SLi (t1)

D H(t)0 V (t)0E = J, L + L (t1)

29 D H(t)0E = J, L (t1)

D H(t)0H(t)E = J, L (t1) D E = J, SLˆ (t1) .

Thus Sˆ is self adjoint on Λ \ V .

Next, define A : V (t) → H(t) by A(t)J(t) = J 0H(t). Then by 3.5 the family Λ/V

satisfies the Riccatti equation

Sˆ0 + Sˆ2 + RH(t) + 3AA∗ = 0. (3.5)

The restriction max{eigenvalue S(α)} ≤ cot α will be shown to also be a restriction on the eigenvalues of Sˆ. Let z ∈ H(t) be a unit vector. Then, since Sˆ is equal to S on H(t),

D E Sˆ(α)z, z = hS(α)z, zi

≤ hcot(α)z, zi

= cot(α).

Hence max{eigenvalue Sˆ(α)} ≤ cot α

Since Λ/V has no zeros on (α, π), by the proof of Theorem 18, Sˆ ≡ cot(t) · Id and

RH(t) ( · , γ˙ )γ ˙ ≡ Id. Then equation 3.5 becomes

− csc2(t) · Id + cot2 ·Id + RH(t) + 3AA∗ = 0

−Id + Id + 3AA∗ = 0

AA∗ = 0.

Since AA∗ is identically zero, for X ∈ H(t) a vector field,

0 = hAA∗X,Xi

30 = hA∗X,A∗Xi and ||A∗X|| = 0 for all X. Therefore A∗ is identically zero. Furthermore, the tensor

A is identically zero which implies that the distribution V (t) is parallel. This means the

distribution H(t) is parallel as well. The only constant curvature 1 Jacobi fields that satisify

max{eigenvalue S(α)} ≤ cot α and have no zeros on (α, π) are sin t · E(t) where E(t) is a parallel field. Thus, for all

J ∈ Λ \ V , J H(t) = sin t · E(t) where E(t) is a parallel field with E (t) ∈ H (t). Also,

J H(t) = sin t · E(t) is itself a Jacobi field as seen by 3.4. Since RH(t) ( · , γ˙ )γ ˙ ≡ Id and

AA∗ ≡ 0

J 00 + RJ + 3AA∗J = − sin(t) · E(t) + sin(t) · E(t) + 0

= 0

and sin t · E(t) is a horizontal Jacobi field.

The requirement that the Riccati operator is self-adjoint on the family Λ is a

3 necessary one. Consider S with the round metric. Let J1,J2 be Jacobi fields. Define

J1(t) := sin(t) · E1(t) + cos(t) · E2(t)

and

J2(t) := cos(t) · E1(t) − sin(t) · E2(t)

where E1(t) and E2(t) are orthogonal parallel fields. Note that

hSJ1,J2i = hcos(t) · E1(t) − sin(t) · E2(t), cos(t) · E1(t) − sin(t) · E2(t)i

31 = cos2(t) + sin2(t)

= 1

yet

hJ1,SJ2i = hsin(t) · E1(t) + cos(t) · E2(t), − sin(t) · E1(t) − cos(t) · E2(t)i

= − sin2(t) − cos2(t)

= −1.

Thus hSJ1,J2i 6= hJ1,SJ2i and S is not self-adjoint. This example meets all of the restric-

tions of the above splitting theorem except for the Riccati equation being self-adjoint. The

conclusion of the theorem is not met as these Jacobi fields have no zeros.

The requirement that the maximum of the eigenvalues be bounded above is also

π necessary. In the above scenario let α = 2 +  and Λ = Ji(t) = cos(t) · Ei(t) where Ei(t) are orthonormal parallel fields. Let  → 0. The Riccati operator S(t) = − tan(t). Thus the maximum of the eigenvalues of S(α) is infinity, yet cot(α) is zero. This example meets all of the restrictions of Theorem 22 except for the restriction on the eigenvalues of S. The conclusion of the theorem is not met as these Jacobi fields have no zeros.

As an application I will reprove the following theorem as stated in [3] using The- orem 22.

Theorem 23 (Bonnet(1855) and Synge(1926)). Suppose (M, g) satisfies sec ≥ k > 0. Then

geodesics with length greater than √π cannot be locally minimizing. k

Proof. Assume for the moment that k = 1. Let γ be a geodesic. Let Λ be an (n − 1)-

dimensional family of Jacobi fields orthogonal to γ with γ(0) = 0. Hence the Riccati

32 operator S is self-adjoint after the additional requirement that J(0) = 0. For use in Theorem

22, let α = 0. Since cot(0) = ∞ there is no restriction on the eigenvalues of S. Then, by

Theorem 22 Λ splits orthogonally into span{J ∈ Λ | J has a zero before time π} ⊕ {J ∈ Λ | where J takes the form sin(t) · E(t)}.

Thus γ has a conjugate point before time π or, given the Jacobi field sin(t)·E(t), a conjugate point at π. Since geodesics only minimize up to their first conjugate point, a geodesic with length greater than π cannot be locally minimizing.

The standard rescaling of manifolds produces the desired result.

33 Bibliography

[1] Manfredo Perdig˜aodo Carmo. Riemannian geometry. Mathematics: Theory & Ap- plications. Birkh¨auserBoston Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty.

[2] Detlef Gromoll and Gerard Walschap. Metric foliations and curvature, volume 268 of Progress in Mathematics. Birkh¨auserVerlag, Basel, 2009.

[3] Peter Petersen. Riemannian geometry, volume 171 of Graduate Texts in Mathematics. Springer, New York, second edition, 2006.

[4] Burkhard Wilking. A duality theorem for Riemannian foliations in nonnegative sectional curvature. Geom. Funct. Anal., 17(4):1297–1320, 2007.

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