Comparison theorem

# Comparison theorem

Discussion

Encyclopedia
A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.

## Riemannian geometry

In Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.
• Rauch comparison theorem
Rauch comparison theorem
In Riemannian geometry, the Rauch comparison theorem is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for large curvature, geodesics tend to converge, while for small curvature,...

relates the sectional curvature
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

of a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

to the rate at which its geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

• Toponogov's theorem
Toponogov's theorem
In the mathematical field of Riemannian geometry, Toponogov's theorem is a triangle comparison theorem. It is one of a family of theorems that quantify the assertion that a pair of geodesics emanating from a point p spread apart more slowly in a region of high curvature than they would in a region...

• 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 http://mathworld.wolfram.com/Berger-KazdanComparisonTheorem.html
• 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 volume comparison theorem / Bishop comparison theorem, conditional on a lower bound for the Ricci curvature
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...

s (R.L. Bishop & R. Crittenden, Geometry of manifolds)
• Lichnerowicz comparison theorem
• Eigenvalue comparison theorem
• Cheng's eigenvalue comparison theorem
Cheng's eigenvalue comparison theorem
In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplaceâ€“Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also...