Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Abstract

Let $M$ be a complete Riemannian manifold possibly with a boundary \pp M. For any $C^1$-vector field $Z$, by using gradient/functional inequalities of the (reflecting) diffusion process generated by L:=\DD+Z, pointwise characterizations are presented for the Bakry-Emery curvature of $L$ and the second fundamental form of \pp M if exists. These extend and strengthen the recent results derived by A. Naber for the uniform norm \|\Ric_Z\|_\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