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Slant submanifolds of Kähler product manifolds. (English) Zbl 1126.53018
In the paper slant submanifolds of a Kähler product manifold are studied. The submanifold $M$ of a Kähler manifold $\overline M$ is said to be slant if for each non zero vector $X\in T_pM$ the angle $\theta(X)$ between $JX$ and $T_pM$ is independent of the choice of $p\in M$ and $X\in T_pM$. Consider a Kähler product manifold $\overline M=\overline M^m\times\overline M^n$. Denote by $\overline P$ and $\overline Q$ the projection operators of the tangent space of $\overline M$ to the tangent spaces of $\overline M^m$ and $\overline{M}^n$, respectively, and put $F = \overline{P}- \overline{Q}$. It is proved that an $F$-invariant, slant submanifold of a Kähler product manifold $\overline M= \overline M^m\times\overline M^n$ is a product manifold $M_1\times M_2$ and $M_1$ (resp., $M_2$) is also a slant submanifold of $\overline M^m$ (resp., $\overline M^n$). It is also obtained that if $M_1\times M_2$ is the Kähler slant submanifold of $\overline M$ then $M_1$ (resp., $M_2$) is a Kähler slant submanifold of $\overline M^m$ (resp., $\overline M^n$). In the last section several inequalities on scalar curvature and Ricci tensor for slant, invariant and anti-invariant submanifolds of a Kähler product manifold $\overline M^m(c_1)\times \overline M^n(c_2)$ are obtained.
53C15Differential geometric structures on manifolds
53C40Global submanifolds (differential geometry)