Pseudo-umbilical spacelike submanifolds in the indefinite space form (Q2770061)

Let \(M^n\) be an \(n\)-dimensional compact spacelike submanifold in an indefinite space form \(M_p^{n+p} (c)\) of index \(p\), which is pseudo-umbilical in the sense that there exists a function \(\lambda\) on \(M^n\) such that \(\langle h(X,Y),\xi \rangle= -\lambda\langle X,Y\rangle\) for the mean curvature vector \(\xi\). For such an \(M^n\) the following integral inequalities are proved NEWLINE\[NEWLINE\int_{ M^n} \Bigl\{\frac 12\sum R^2_{mijk}+ \sum R^2_{mj}-ncR+ nH^2r\Bigr\}*1 \leq 0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\int_{M^n}\Bigr\{ \frac 12\sum R^2_{mijk}+ (n-2)cS- n(n-1) c^2+n^2(n-1)cH^2+n H^2S\Bigr\} *1\leq 0,NEWLINE\]NEWLINE where \(S\) is the square length of the second fundamental form \(h\) and \(H\) is the mean curvature of \(M^n\). For the second inequality is proved that equality holds if and only if \(M^n\) is totally geodesic.