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UC Riverside UC Riverside Electronic Theses and Dissertations UC Riverside UC Riverside Electronic Theses and Dissertations Title A Jacobi Field Splitting Theorem for Positive Curvature Permalink https://escholarship.org/uc/item/9hp96500 Author Gumaer, Dennis Publication Date 2013 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA RIVERSIDE A Jacobi Field Splitting Theorem for Positive Curvature 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 sectional curvature. Theorem 1. Let M be an n-dimensional Riemannian manifold with sec ≥ 1. For α 2 [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 maxfeigenvalue S(α)g ≤ cot α then Λ splits orthogonally into spanfJ 2 Λ j J has a zero before time πg ⊕ fJ 2 Λ j where J takes the form sin(t) · E(t)g 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 geodesics 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 Riemannian geometry. 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 2 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. jjvjj = 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 jjγ_ jj = 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 :(−"; ") × [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 :(−"; ") × [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 covariant derivative r·· 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 ! rX on (M; g) is uniquely defined by the following properties: • Y ! rY X is a (1,1)-tensor i.e.
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