Comparison theorem

Comparison theorem

A comparison theorem is any of a variety of 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 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 [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
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
  • Zeeman's comparison theorem, a technical tool from the theory of spectral sequences



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