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

摘要

We establish variational formulas for Ricci upper and lower bounds, as wellas a derivative formula for the Ricci curvature. As applications, constantcurvature manifolds, Einstein manifolds and Ricci parallel manifolds areidentified, respectively, with different integral-differential formulas andsemigroup inequalities. Moreover, by using derivative and Hessian formulas forthe heat semigroup $P_t$ developed from stochastic analysis, explicit Hessianestimates are derived on Einstein and Ricci parallel manifolds.