Totally real pseudo-umbilical submanifolds of a complex projective space (Q1382851)

Let \(M\) be an \(n\)-dimensional totally real submanifold of complex projective space \(\mathbb{C} P^n\) and \(\zeta\) the mean curvature vector field of \(M\). If the second fundamental form \(h\) of \(M\) satisfies \(\langle h(X, Y),\zeta\rangle= \lambda\langle X,Y\rangle\) for all tangent vectors \(X\), \(Y\) of \(M\) and for some function \(\lambda\) on \(M\), then \(M\) is called pseudo-umbilical. The author derives some integral formulae for the mean curvature of \(M\) and the length of \(h\) in the case that \(M\) is compact, totally real and pseudo-umbilical. From these integral formulae some inequalities are derived which are sufficient in order that a compact totally real pseudo-umbilical submanifold of \(\mathbb{C} P^n\) is totally geodesic or a 2-torus.