Comparison theorem

Comparison theorem is a popular name for theorems that compare properties of various mathematical objects

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 allows to compare various geometric quantities in a Riemannian manifold with fixed upper bound on all sectional curvatures to those in a simply connected manifold of constant sectional curvature k, which is unique up to isometry, i.e., taking scaling into an account, eventually referring to the model spaces, which are hyperbolic, Euclidean and spherical n-spaces.
• Zeeman comparison theorem (Zeeman's comparison theorem)
• Hessian comparison theorem
• Laplacian comparison theorem
• Morse-Schoenberg comparison theorem
• Berger comparison theorem, Raush-Berger comparison theorem, M.Berger, "An Extension of Raush's Metric Comparison Theorem and some Applications", Jllinois J. Math., vol. 6 (1962) 700-712
• Berger-Kazdan comparison theorem [1]
• 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 curvatures (R.L. Bishop & R. Crittenden, Geometry of manifolds)
• Lichnerowicz comparison theorem
• Eigenvalue comparison theorem
  • Cheng's eigenvalue comparison theorem

See also: Comparison triangle

Differential equations

In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof) provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property. See also Lyapunov comparison principle

• Sturm comparison theorem [2]

Other

• Limit comparison theorem, about convergence of series
• Comparison theorem for integrals, about convergence of integrals