Ricci curvature and volume growth (Q1814489)

The well known Bishop-Gromov inequality for the volume growth of balls in a Riemannian manifold \(M\) has an analogue for tubes around a compact totally geodesic submanifold \(L\subset M\), but instead of a lower Ricci curvature bound one has to assume a lower bound \(\kappa\) for the radial sectional curvatures of \(M\) with respect to \(L\). The comparison space is the normal bundle of \(L\) with a metric with radial curvatures \(\equiv \kappa\). The present paper shows by a counterexample that in fact a Ricci curvature bound is not sufficient in this case. More precisely, the authors construct a metric with nonnegative Ricci curvature on \(M\) = Hopf bundle over \(\mathbb{C} P^ 1\) such that the zero section \(L=\mathbb{C} P^ 1\) is totally geodesic, and the volume ratio of \(r\)-tubes around \(L\) in \(M\) and the comparison space goes to \(\infty\) as \(r\to \infty\) (instead of being monotonically decreasing). In a previous work, the authors have shown that in the equality case, the normal bundle of \(L\) must be flat provided that all sectional curvatures in \(M\) are \(\geq \kappa\). A second example in the present paper (with \(M=\mathbb{C} P^ 2\), \(L=\mathbb{C} P^ 1\)) shows that for this result it does not suffice to have only the radial sectional curvatures and the normed Ricci curvature bounded by \(\kappa\). The Ricci curvature computations are explicit and in the spirit of Sha/Yang or Besse, p. 274.