ON RIEMANNIAN MANIFOLDS WITH POSITIVE WEIGHTED RICCI CURVATURE OF NEGATIVE EFFECTIVE DIMENSION

Abstract

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound Ric_N ≥ K with K > 0 for the negative effective dimension N < 0. We analyze two one-dimensional examples of constant curvature Ric_N ≡ K with finite and infinite total volumes. We also discuss when the first non-zero eigenvalue of the Laplacian takes its minimum under the same condition Ric_N ≥ K > 0, as a counterpart to the classical Obata rigidity theorem. Our main theorem shows that, if N < −1 and the minimum is attained, then the manifold splits off the real line as a warped product of hyperbolic nature.