Comparison Theorem - Riemannian Geometry

Riemannian Geometry

In Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.

• Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart.
• Toponogov's theorem
• Myers's theorem
• Hessian comparison theorem
• Laplacian comparison theorem
• Morse–Schoenberg comparison theorem
• Berger comparison theorem, Rauch–Berger comparison theorem, M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", llinois J. Math., vol. 6 (1962) 700–712
• Berger–Kazdan comparison theorem
• Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold) F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356).
• Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures (R.L. Bishop & R. Crittenden, Geometry of manifolds)
• Lichnerowicz comparison theorem
• Eigenvalue comparison theorem
  • Cheng's eigenvalue comparison theorem

See also: Comparison triangle