Pointwise Characterizations of Curvature and Second Fundamental Form on Riemannian Manifolds

Abstract

Let $M$ be a complete Riemannian manifold possibly with a boundary $\partial M$. For any $C^1$-vector field $Z$, by using gradient/functional inequalities of the (reflecting) diffusion process generated by $L:=\Delta +Z$, pointwise characterizations are presented for the Bakry-Emery curvature of $L$ and the second fundamental form of $\partial M$ if exists. These extend and strengthen the recent results derived by A. Naber for the uniform norm $\Vert {\mathbf{Ric}}_Z{\Vert }_{\infty }$ on manifolds without boundary. A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first named author, such that the proofs are significantly simplified.

References in corpus (1)

• Remarks on spectral gaps on the Riemannian path space

Cited by in corpus (3)

• Characterizations of the upper bound of Bakry-Emery curvature
• Spectral gap on Riemannian path space over static and evolving manifolds
• Characterization of pinched Ricci curvature by functional inequalities