THE SPHERE THEOREM IN RIEMANNIAN GEOMETRY

THOMAS YU

Abstract. We will give a proof of the Sphere Theorem in Riemannian Ge- ometry, which states that if M is a complete, simply-connected n-dimensional 1 manifold with sectional curvature KM such that 1 ≥ KM > 4 , then M is homeomorphic to the n-sphere Sn

Contents 1. Notation and Fundamental Notions 1 2. Some Necessary Lemmas 5 3. Proof of the Sphere Theorem 6 4. Acknowledgements 7 References 7

1. Notation and Fundamental Notions For the sake of brevity, we will assume that the reader has completed a course in differential geometry. The proof of the Sphere Theorem given is taken largely from Comparison Theorems in Riemannian Geometry by Cheeger and Ebins; since this theorem requires 5 fairly dense chapters of background material, in this section we present without proof an overview of the most significant theorems and construc- tions used in the proof of the Sphere Theorem. Throughout the paper, assume M is a smooth, finite-dimensional manifold. For p ∈ M, let Mp denote the tangent space to M at p, let ζ(M) denote the linear space of smooth vector fields on M, and let F(M) denote the ring of smooth, real-valued functions on M. Recall that tangent vectors are derivations, and we interpret them as giving directional derivatives. Definition 1.1. A Riemannian metric is an assigment to each p ∈ M of a sym- metric, positive-definite bi-linear form h,ip on Mp such that for any smooth vector fields V,W , the function p → hV,W ip is a smooth real-valued function on M. A manifold that possesses a Riemannian metric is called a Riemannian manifold. Note that this bi-linear form defines a norm on the tangent space; we shall denote 1 hV,V i 2 as kV k. Definition 1.2. An affine connecton is a connection on the tangent bundle T(M). We can view an affine connection as a bi-linear map ∇ : ζ(M) × ζ(M) → ζ(M)(1.3)

Date: August 12, 2013. 1 2 THOMAS YU where ζ(M) is the linear space of smooth vector fields on M, which obeys the following properties:

(1.4) ∇fV W = f∇V W

(1.5) ∇V fW = (V f)W + f∇V W

∀f ∈ F(M),V,W ∈ ζ(M). Affine connections provide a natural way of differentiat- ing arbitrary tensor fields on M since a connection on T (M) induces connections on arbitary tensor products of the tangent and co-tangent bundles. For this paper we will only require differentiating vector fields; ∇V W is called the covariant derivative of W in the direction V . We now state the fundamental theorem of Riemannian geometry.

Theorem 1.6. For every Riemannian metric there is a unique affine connection called the Levi-Civita connection that satisfies the following properties

(1.7) X hV,W i = h∇X V,W i + hV, ∇X W i

∇V W − ∇W V − [V,W ] = 0(1.8) where [, ] denotes the Lie bracket [X,Y ]f = (XY − YX)f.

The first condition in the theorem describes the compatibility of the connection and the metric; that is, norms of vectors transported along curves are preserved. The second condition is equivalent to saying that the connection is torsion-free. From here on, ∇ will refer to the Levi-Civita connection. With the Riemannian metric we can define the length of a smooth curve c : [a, b] → M,L[c] as

Z b (1.9) L[c] = kc0(t)kdt. a Further, we can make M into a metric space by defining the distance between two points p, q ∈ M as

(1.10) ρ(p, q) = infc∈C L[c]. where C is the set of all curves between p, q. We will now explore some conditions for the existence of a curve γ from p to q such that L[γ] = ρ(p, q). Readers familiar with the calculus of variations will recognize the following condition. For the following discussion assume that curves c are smooth, parametrized by arc length, and kc0(t)k = l, where l is some non-zero constant.

Definition 1.11. Let α :[a, b] × [−, ] → M be a smooth function such that α|[a,b]×0 = c :[a, b] → M. Then α is a called a smooth variation of c. Intuitively, α encodes perturbations of the curve c; that is, we can think of α as a family of curves parametrized by s ∈ [−, ]. Now let T,V be the tangent vectors on [a, b] × [−, ] corresponding to the first and second variables respectively; note that T,V are just the derivative operators with respect to the associated variable. We now compute the change in arc length over the family of curves cs = α|[a,b]×s THE SPHERE THEOREM IN RIEMANNIAN GEOMETRY 3 around the base curve c0. This is given by b b d d Z 1 Z 1 0 0 2 2 (1.12) L[cs] = hcs(t), cs(t)i dt = V hT,T i dt ds ds a a b 1 Z −1 (1.13) = hT,T i 2 V hT,T i dt 2 a b Z −1 2 (1.14) = hT,T i h∇V T,T i dt a Note that we used the properties of the Levi-Civita connection above. Since partial derivatives commute in the Euclidean rectangle [a, b]×[−, ], we can rewrite (1.14) as b Z −1 2 (1.15) hT,T i h∇T V,T i dt a 0 Using the kc0k = l, we have that (1.16) Z b d −1 L[cs]|s=0 = l h∇T V,T i dt ds a Z b Z b ! −1 −1 b (1.17) = l (T hV,T i − hV, ∇T T i)dt = l hV,T i |a − hV, ∇T T i dt a a (1.17) is called the first variation formula. Note that if the family of curves share the same endpoints then the first term of (1.17) vanishes. Further, if c0 is the d shortest curve between the endpoints, we must have that ds L[cs]|s=0 = 0 for any α. Thus, any vector field V which vanishes at the endpoints of c0 must satisfy the last term of (1.17) being 0. By choosing V = φ(t)∇T T where φ(t) > 0 for a < t < b and φ(t) = 0 for t = a, b, we see that

∇T T = 0(1.18) This motivates the following definition of a geodesic curve.

0 0 Definition 1.19. A smooth curve is called a geodesic if ∇cc = 0. A geodesic is normal if kc0k = 1. Note that the defining property of a geodesic, written in a coordinate neighbor- hood is a second order differential equation in the parameter t of the geodesic; to see this, we simply need to translate the vector field c0 as well as the connection in terms of differential operators in the chosen coordinate basis. Hence, from existence and uniqueness theorems from the theory of ordinary differential equations, the speci- fication of a point p ∈ M and v ∈ Mp results in a unique geodesic γv through p whose tangent vector at p is v. We are now ready to define the exponential map.

Definition 1.20. The exponential map exp(v), is defined by γv(1), for v ∈ Mp such that 1 is in the domain of γv.

expp is defined in a neighborhood of the origin of Mp, is smooth, and is a local diffeomorphism in a neighborhood of the origin by the Inverse Function Theorem since the differential of the exponential map is nonzero at the origin. An interesting theorem concerning the exponential map is the Hopf-Rinow Theorem 4 THOMAS YU

Theorem 1.21. A connected, Riemannian manifold M is complete as a metric space with respect to the metric described above if and only if the exponential map is defined over all of Mp for every p ∈ M. In particular, this implies that in a complete manifold, there is always a length minimizing geodesic between any two points in the manifold. We know that for each p ∈ M, expp is injective on a sufficiently small ball; therefore, ρ(γ(t), γ(0)) = t for sufficiently small t(if we parametrize γ by arc length). A question that arises is how large t can be before this formula no longer holds. Definition 1.22. The injectivity radius i(M) of M is the supremum over all r such that ∀p ∈ M, expp is an embedding on the open ball of radius r in Mp

Definition 1.23. Define d(M) to be supp,q∈M ρ(p, q).

We now define the curvature tensor which is a trilinear map R : Mp ×Mp ×Mp → Mp.

(1.24) R(X,Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z We can think of the curvature as measuring the failure of commutativity of covariant differentiation.

Definition 1.25. Given a plane σ in Mp and two spanning vectors v, w, we define the sectional curvature K(σ) as hR(v, w)w, vi (1.26) K(σ) = kv ∧ wk2 where ∧ is the wedge product and kv ∧ wk2 denotes the square of the area of the parallelogram formed by v, w. It can be shown that K(σ) is independent of the choice of v, w.

The notation KM > H denotes that for all plane sections at all points of M, the sectional curvature is bigger than the constant H. Definition 1.27. We define M H to be the simply-connected 2-dimensional space of constant curvature H. The use of M is to emphasize its connection to M through a theorem we state below. Having constructed all the necessary tools to understand the statement of the Sphere Theorem, we will now state some theorems that are used in the proof of the sphere theorem. We now state two equivalent formulations of Toponogov’s Theorem

Theorem 1.28. Let M be a complete manifold with KM ≥ H then 1. Let(γ1, γ2, γ3) determine a geodesic triangle. Suppose γ1, γ3 are minimal and if H > 0, L[γ ] ≤ √π . Then in the simply-connected 2-dimensional space of constant 2 H H curvature H (denote as M ) there exists a geodesic triangle (γ1, γ2, γ3) such that L[γi] = L[γi] and α1 ≤ α1, α3 ≤ α3. 0 0 2. Let γ1, γ2 be geodesic segments in M such that γ1(l1) = γ2(0) and ∠(−γ1(l2), γ2(0)) = α. Let γ be minimal and if H > 0, L[γ ] ≤ √π . Let γ , γ in M H be such that 1 2 H 1 2 0 0 γ1(l1) = γ2(0),L[γi] = Lγi=li, and ∠(γ1 (l2), γ2 (0)) = α. Then

(1.29) ρ(γ1(0), γ2(l2)) ≤ ρ(γ1(0), γ2(l2)). THE SPHERE THEOREM IN RIEMANNIAN GEOMETRY 5

Definition 1.30. In the future, we will refer to (γ1, γ2, α) with γ1, γ2, α as in statement 2 of the theorem above as a hinge. We now state a useful theorem that relates the sectional curvature of M to the injectivity radius i(M). Theorem 1.31. Let M be a simply-connected, complete Riemannian manifold such that K ≥ K ≥ H > K > 0 for some K ∈ . Then i(M) ≥ √π . M 4 R K We are now finished with the preliminaries; in the next section, we will prove several lemmas using the preliminaries that directly lead to the Sphere Theorem.

2. Some Necessary Lemmas Throughout this section let M be a compact, simply-connected, n-dimensional Riemannian manifold; further, assume all geodesics are parametrized by arc length. Lemma 2.1. Fix p ∈ M and choose q ∈ M such that ρ(p, q) is maximal. Then for any vector v ∈ Tq, there exists a minimal geodesic γ from q to p such that 0 π ∠(γ (0), v) ≤ 2 . Proof. From our section on fundamental notions, we know there is a unique σ that 0 is the geodesic from q such that σ (0) = v. Pick a sequence of points along σ {qi} so that {qi} → q, and let γi be the minimal geodesic from qi to p. Suppose there 0 π is a subsequence γj such that ∠(γj(0), v) ≤ 2 . This subsequence must converge to 0 π some minimal geodesic γ and by continuity of the bi-linear form, ∠(γ (0), v) ≤ 2 . Hence, we may assume for a contradiction that ∃t0 > 0 such that ∀t ∈ (0, t0] and 0 0 π for any minimal geodesic γt from p to σ(t), we have ∠(γt, σ ) > 2 . Now recall our discussion of the first variation of arc length. Consider a variation ds of γt such that ds goes from p to σ(t − s). We see that the first variation formula reduces to d (L[d ])| = hγ0, σ0i < 0(2.2) ds s s=0 t −1 since in our case l = 1 and ∇T T = 0 since T is the vector field of the geodesic π γt. Note that above inequality follows from cos(x) < 0 for 2 < x < π. Hence as t goes from t0 to 0, we see that the distance between p and σ(t) is strictly decreasing. This is a contradiction since by assumption, p and q are maximally separated. The lemma then follows.  Lemma 2.3. Let K ≥ H > 0 and d(M) > √π . Then for p, q as in Lemma 2.1, M 2 H M = B √π (p) ∪ B √π (q). 2 H 2 H Proof. The inclusion of the neighborhoods around p, q in M is clear so we only need to show that M is contained in those neighborhoods. Let q1 ∈ M be any point such that l = ρ(p, q ) > √π , and let γ be a minimal, normal geodesic from p = γ (0) 1 1 2 H 1 1 to q1 = γ1(l1). By Lemma 2.1, we know there is a minimal, normal geodesic γ2 from 0 0 π p = γ2(0) to q = γ2(l) such that α = ∠(γ1(0), γ2(0)) ≤ 2 . We now apply the second formulation of Toponogov’s theorem (1.29) to the hinge defined by (γ1, γ2, α). Let n (γ1, γ2, α) be a comparison hinge in S , with corresponding points q1, q, p. From Theorem 1.29 we know that

(2.4) ρ(q1, q) ≤ ρ(q1, q). 6 THOMAS YU

Applying the spherical law of cosines we have that √ √ cos( Hρ(q1, q)) ≥ cos( Hρ(q1, q))(2.5) √ √ √ √ (2.6) = cos( Hl) cos( Hl1) + sin( Hl) sin( Hl1) cos(α) √ √ ≥ cos( Hl) cos( Hl1) > 0(2.7) It then follows that ρ(q , q) < √π . Thus given any point x ∈ M, x must be in 1 2 H either B √π (p) or B √π (q), which completes the lemma. 2 H 2 H  1 Lemma 2.8. Let 1 ≥ KM ≥ H > 4 , ρ(p, q) be maximal. Then for all geodesics γ that start from p there is a unique r along γ such that ρ(p, r) = ρ(q, r) ≤ √π . 2 H Proof. From Theorem 1.31 we have that i(M) ≥ π ≥ √π . Thus, ρ(p, γ(t)) = t for 2 H 0 ≤ t ≤ π. By the lemma above, ρ(q, γ(π)) ≤ √π . Note that ρ(q, γ(t)) − ρ(p, γ(t)) 2 H is a continuous real valued function which is positive at t = 0 and negative at t = π; hence, by the Intermediate Value Theorem, there exists tr such that γ(tr) = r is a point that satisfies the conditions above. So existence is established. Now assume that there exist r1, r2 along γ that are both equidistant from p, q. Assume ρ(p, r1) < ρ(p, r2). Then we have that

ρ(q, r2) = ρ(p, r2) = ρ(p, r1) + ρ(r1, r2) = ρ(q, r1) + ρ(r1, r2)(2.9)

Let σ be a minimal geodesic from r1 to q; then the segment of −γ from r2 to r1 concatenated with σ should form a smooth geodesic from r2 to q, so σ must coincide with the segment −γ from r1 to p. This implies that p = q, which is a contradiction. Thus r is unique. 

3. Proof of the Sphere Theorem Theorem 3.1. If M is a complete, simply-connected n-dimensional manifold with 1 sectional curvature KM such that 1 ≥ KM > 4 , then M is homeomorphic to the n-sphere Sn. Proof. We will exhibit an explicit homeomorphism between M and Sn with the 1 help of the lemmas in the preceding section. Suppose 1 ≥ KM ≥ H > 4 . Let p, q ∈ M be at maximal distance. Let p, q ∈ Sn denote antipodal points on the unit n n n n-sphere and I : Sp → Mp an isometry. For each nonzero vector v ∈ Mp define a n n map f : Mp → Mp , f(v) = t0v by letting exp(f(v)) be the point along the geodesic t → expp tv which is equidistant from p and q. The existence of f(v) satisfying π f(v) ≤ √ < π ≤ i(M)(3.2) 2 H follows from Lemma 2.8. We now define a map h : Sn → M by  p : x = p  2ρ(x,p) −1  exp ( (f ◦ I ◦ exp (x))) : x ∈ B π (p) − p p π p 2 (3.3) h(x) = 2ρ(x,q) −1 −1 exp ( (exp ◦ exp ◦I ◦ exp (x))) : x ∈ B π (q) − q  q π q p p 2  q : x = q By inspection of h, we see that failure of continuity could only result from the map v → kf(v)k. Continuity of this map follows from the continuity of the distance function coupled with the uniqueness of f(v) supplied by Lemma 2.8. Thus, h THE SPHERE THEOREM IN RIEMANNIAN GEOMETRY 7 is continuous. Since kf(v)k < i(M), we see that h|B π (p), h|B π (q) are injective. 2 2 Hence to show the injectivity of h it suffices to show that

(3.4) h(B π (p)) ∩ h(B π (q)) = ∅. 2 2 Now, suppose that x ∈ h(B π (p)). Then x = γ(t) with γ(0) = p and ρ(x, p) < 2 kf(γ0(0)k). Lemma 2.8 coupled with the Intermediate Value Theorem show that ρ(x, p) < ρ(x, q). By the same argument, if x ∈ h(B π (q)), then ρ(x, q) < ρ(x, p). 2 Thus we see that no point x can line in the intersection of the images of the two balls; hence, h is injective. If ρ(x, p) ≤ ρ(x, q), let γ be a minimal normal geodesic from p to x such that x = γ(t). By Lemma 2.8 and the Intermediate Value Theorem, ∃t ≥ t such that t = kf(v)k. Then it follows that x ∈ h(B π (p)). o 0 2 By symmetry, if ρ(x, q) ≤ ρ(x, p) then x ∈ h(B π (q)). Hence, h is surjective. Since 2 h is a injective, surjective, continuous map from a compact space to a Hausdorff space, h is a homeomorphism.  Remark 3.5. A natural question that arises is whether under the same conditions, M is diffeomorphic to Sn. The existence of exotic spheres, manifolds that are homeomorphic but not diffeomorphic to spheres, makes this question non-trivial. As it turns out, the Differential Sphere Theorem was proved rather recently in 2007 by Brendle-Schoen. The statement of the theorem is identical to our statement, except homeomorphic is replaced with diffeomorphic. In addition, one can ask whether the strict inequality in the lower bound on the n sectional curvature KM is necessary. The complex projective spaces CP equipped with the Fubini-Study metric is a compact, simply-connected Riemannian manifold 1 n whose sectional curvature K obeys 1 ≥ K ≥ 4 but CP is not homeomorphic to S2n. Thus we see that the strict inequality in the lower bound in the statement of the Sphere Theorem is necessary.

4. Acknowledgements I would like to thank Peter May and the University of Chicago Math Department for providing me with monetary and pedagogical support in learning new math over the summer. I would like to extend my sincere gratitude toward my mentor Jonathan Gleason who very patiently and expertly explained mathematical and physical principles to me and also endured at least three sweeping changes in the topic for my paper.

References Chern S.S. Chern, W.H. Chen, K.S. Lam, Lectures on Differential Geometry, World Scientific Publishing, Singapore, 2nd Edition, 2000. Jeff Cheeger, David G. Ebin, Comparison Theorems in Riemannian Geometry, North Holland Publishing Company, Amsterdam, 1975. Zuoqin Wang, Notes on Differential Geometry, Lecture 36: The Sphere Theorem 2012