The Bakry–Emery Ricci tensor and its applications to some compactness theorems

Abstract

Let (M, g) be a complete and connected Riemannian manifold of dimension n. By using the Bakry-Emery Ricci curvature tensor on M, we prove two theorems which correspond to the Myers compactness theorem.

Manifolds with Bakry-Emery Ricci Curvature Bounded Below

“…In this paper we establish a Myers type theorem for manifolds bounded below by a negative constant. Therefore we prove that is a generalization of the theorem of M. Limoncu in [2] or H. Tadano in [3].…”

Integral curvature bounds and bounded diameter with Bakry–Emery Ricci tensor

“…Several works have attempted to generalize this result (for example, see [4], [5], [9], and [10]), including that of Sprouse which can be summarized in the following three theorems.…”

Comparison Geometry for Integral Bakry–Émery Ricci Tensor Bounds

“…Petersen and Sprouse [17] have proved the case when f is constant. For other Myers' type theorems on smooth metric measure spaces, see [22,15,20].…”