Identifying Constant Curvature Manifolds, Einstein Manifolds, and Ricci Parallel Manifolds

Abstract

We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. Combining these with derivative and Hessian formulas of the heat semigroup developed from stochastic analysis, we identify constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds by using analytic formulas and semigroup inequalities. Moreover, explicit Hessian estimates are derived for the heat semigroup on Einstein and Ricci parallel manifolds.