Second fundamental form and gradient of Neumann semigroups

The second fundamental form for the boundary of a compact Riemannian manifold is described by a short time behavior for the gradient of Neumann semigroups. As an application, we prove that the manifold is convex if and only if the Neumann heat semigroup Pt satisfies the gradient estimate |∇Ptf|⩽eKtPt|∇f| for some constant K∈R and all t⩾0, f∈C1.