Characterizations of real hypersurfaces of a complex hyperbolic space in terms of Ricci tensor and holomorphic distribution (Q1345354)

Let \(M\) be a real hypersurface of the complex hyperbolic space \({\mathbb{C}}H^ n\) \((n \geq 2)\) equipped with the Fubini Study metric of constant holomorphic sectional curvature \(-4\). Denote by \((\phi, \xi,\eta, g)\) the induced almost contact metric structure, by \(S\) the Ricci tensor and by \(A\) the shape operator of \(M\), and put \(h = \text{trace }A\). Let \(T^ 0 M\) be the orthogonal complement of the span of \(\xi\) in \(TM\), \(\nabla^ 0\) the induced connection on \(T^ 0 M\) and \(A^ 0 = A \mid T^ 0 M\). The author proves the following two results: (1) \(\| \nabla S\|^ 2 \geq \{2n (h-\eta(A\xi)) + (\phi A \xi) \cdot h - \text{trace}(\nabla_ \xi A)A\phi)\}^ 2/(n-1)\), and equality holds if and only if \(M\) is locally congruent to a horosphere in \({\mathbb{C}}H^ n\), or to a geodesic hypersphere in \({\mathbb{C}}H^ n\), or to a cube around some totally geodesic \({\mathbb{C}}H^{n-1} \subset {\mathbb{C}}H^ n\); (2) If \(\nabla^ 0_ X A^ 0 = 0\) for all vectors \(X\) in \(T^ 0 M\), then \(M\) is locally congruent to a horosphere in \({\mathbb{C}}H^ n\), or to a tube around some totally geodesic \({\mathbb{C}}H^ k \subset {\mathbb{C}}H^ n\) for some \(k \in \{0,\dots, n-1\}\), or to a tube around the totally geodesic \({\mathbb{R}}H^ n \subset {\mathbb{C}}H^ n\), or to a ruled real hypersurface in \({\mathbb{C}}H^ n\).