Semi-Riemannian Geometry

Chapter 21 Riemannian Manifolds

In this chapter, we narrow the scope from semi‐Riemannian manifolds to Rie‐ mannian manifolds and examine selected earlier topics in that more specialized setting.

21.1 Geodesics and Curvature on Riemannian Manifolds

Here is the promised generalization of Theorem 19.9.3 to Riemannian manifolds.

Theorem 21.1.1 (Length Minimizing Property of Geodesics on Riemannian Manifolds)

Let be a Riemannian manifold, let p be a point in M, and let V be a normal neighborhood of p in M. Let q be a point in V, let , and let ρ _{p, v}  : [0, 1] → M be the radial geodesic with starting point p and initial velocity v. If λ(s) : [0, 1] → V is a smooth curve such that λ(0) = p,  λ(1) = q , and λ([0, 1]) ⊂ V , then L(ρ _{p, v} ) ≤ L(λ).

Proof

We have from Theorem 18.9.6 that ρ _{p, v} is the unique radial geodesic such that ρ _{p, v} (0) = p,  ρ _{p, v} (1) = q , and ρ _{p, v} ([0, 1]) ⊂ V . By Theorem 19.9.1 (a), ρ _{p, v} has constant speed

so (19.2.1) gives

(21.1.1)

Let be the diffeomorphism corresponding to V, and consider ...