Improved Chen-Ricci inequality for curvature-like tensors and its applications (Q719865)

The author proves several results on a Chen-Ricci inequality involving Ricci curvature and the squared mean curvature for a submanifold \(M\) of ambient manifold \(\widetilde M\). Contents: Chen-Ricci inequality, improved Chen-Ricci inequality, Lagrangian submanifolds, Kählerian slant submanifolds, \(C\)-totally real submanifolds. Let \(\sigma\) be second fundamental form of the immersion \(M\) into \(\widetilde M\), and \(H\) is mean curvature of \(M\) in \(\widetilde M\). \(\text{Rec}(X)\) denote the Ricci curvature of \(M\) in \(X\in T^1_p M\), and \(T^1_p M\) is the set of unit vectors in tangent space \(T_p M\), \(p\in M\). The main result is as follows: Theorem. ``Let \(M\) be an \(n\)-dimensional submanifold of a Riemannian manifold. Then, the following statements are true. (a) For \(X\in T^1_p M\), it follows that \[ \text{Ric}(X)\leq {1\over 4} n^2\| H\|^2+ \widetilde{\text{Ric}}_{(T_p M)}(X),\tag{\(*\)} \] where \(\widetilde{\text{Ric}}_{(T_p M)}(X)\) is the \(n\)-Ricci curvature of \(T_p M\) at \(X\in T^1_p M\) with respect to the ambient manifold \(\widetilde M\). (b) The equality case of \((*)\) is satisfied by \(X\in T^1_p M\) if and only if \(\sigma(x,y)= \theta\) for all \(y\in T_p M\) orthogonal to \(X\), \(2\sigma(X,X)= nH(p)\). If \(H(p)= 0\), then \(X\in T^1_p M\) satisfies the equality case of \((*)\) if and only if \(X\in N(P)\). (c) The equality case of \((*)\) holds for all \(X\in T^1_p M\) if and only if either \(p\) is a geodesic point or \(n= 2\) and \(p\) is an umbilical point.'' Here \((*)\) denote the inequality \(| H\|^2\geq {4\over n^2}\{\text{Ric}(X)-(n- 1)c\}\). The author also studies Chen-Ricci inequality for submanifolds in contact metric manifolds. the exposition is intelligible.