Integral Radial m-Bakry–Émery Ricci Curvatures, Riccati Inequalities, and Ambrose-type Theorems

Inspired by a recent work due to J.-Y. Wu (Potential Anal 58:203–223, 2023), we prove several new compactness criteria for complete Riemannian manifolds via integral radial m-Bakry–Émery Ricci curvatures when m is a positive constant, a negative constant, or infinity. Our results not only generalize the classical compactness criterion via Ricci curvature due to W. Ambrose (Duke Math. J. 24:345–348, 1957), but also generalize a compactness criterion via integral radial Bakry–Émery Ricci curvature due to J.-Y. Wu (Potential Anal 58:203–223, 2023). The key ingredients in proving our results are Riccati inequalities obtained from Bochner–Weitzenböck formulas via m-Bakry–Émery Ricci curvatures and suitable choices of growth conditions on integral radial m-Bakry–Émery Ricci curvatures, potential functions, and norms of potential vector fields.