Lecture 38: Applications of Critical Point Theory

1. Generalized Sphere Theorem As a first application of the critical point theory of distance funciton, we shall prove Theorem 1.1 (Grove-Shiohama). Let (M, g) be a complete simply connected Riemann-ian manifold with K > 1 4 and diam(M, g) ≥ π, then M is homeomorphic to S m. So Grove-Shiohama's theorem implies the sphere theorem. Proof. Let p, q ∈ M so that dist(p, q) = diam(M, g). In what follows we will prove that p has no other critical points. So from the Reeb type theorem we proved last time, M is homeomorphic to S m. Suppose to the contrary, ¯ q = q is a critical point of p. Let γ be a minimal geodesic from q = γ(0) to ¯ q = γ(l). By definition of critical points, there exists a minimal geodesic σ from ¯ q = σ(0) to p = σ(l) so that α = ∠(− ˙ γ(l), ˙ σ(0)) ≤ π 2. Similarly, there exists minimal geodesics γ 1 , σ 1 from p = γ 1 (0) = σ 1 (0) to q = γ 1 (l) = σ 1 (l) so that β = ∠(− ˙ σ(l), ˙ σ 1 (0)) ≤ π 2 , β = ∠(− ˙ γ 1 (l), ˙ γ(0)) ≤ π 2. Since M is compact, there exists k > 1 4 so that K ≥ k. According to Toporogov comparison theorem (triangle version), there is a geodesic triangle in S m (1 √ k) whose sides have length l, l , l while all three angles˜α, ˜ β, ˜ β are all no more than π 2. Since l = dist(p, q) ≥ π, the cosine law in S m (1 √ k) 0 > cos(√ kl) = cos(√ kl) cos(√ kl) + sin(√ kl) sin(√ kl) cos(˜ α) implies exactly one of l and l , say l , is strictly greater than π 2 √ k , and the other one, l, is strictly smaller than π 2 √ k. It follows that cos(√ kl) = cos(√ kl) cos(√ kl) + sin(√ kl) sin(√ kl) cos(˜ α) > cos(√ kl).