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 C1-vector field Z, by using gradient/functional inequalities of the (reflecting) diffusion process generated by L:= Δ+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 ||Ric Z || 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.


curvature second fundamental form diffusion process path space 


58J60 58J58 


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This work was supported by National Natural Science Foundation of China (Grant Nos. 11771326 and 11431014). The authors thank the referees for the helpful comments


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Applied MathematicsTianjin UniversityTianjinChina
  2. 2.Department of MathematicsSwansea UniversitySingleton ParkUK
  3. 3.School of Mathematical SciencesFudan UniversityShanghaiChina

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