2024年3月14日发(作者：绿巨人入口app黑科技天堂网)

比较定理 the comparison theorem

A Note on Comparison Theorems for Nonnegative Matrices

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 [2]



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

Lyapunov comparison theorem

qualitative analysis of large scale dynamical systems