Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Let M be a complete Riemannian manifold possibly with a boundary?M.For any C;-vector field Z,by using gradient/functional inequalities of the（reflecting）diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.