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Semi-slant submanifolds of a locally product manifold. (English) Zbl 1093.53025

Let $\left(\stackrel{˜}{M},g,F\right)$ be a ${C}^{\infty }$ locally product Riemannian manifold, where $g$ is a Riemannian metric and $F$ is a non-trivial tensor field of type (1,1) satisfying the following conditions:

${F}^{2}=I\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}g\left(FX,FY\right)=g\left(X,Y\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\stackrel{˜}{\nabla }F=0\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}X,Y\in T\stackrel{˜}{M}\phantom{\rule{0.166667em}{0ex}},$

$\stackrel{˜}{\nabla }$ being the Levi-Civita connection on $\stackrel{˜}{M}$.

The authors study slant, bi-slant and semi-slant submanifolds of a locally product manifold.

Let $M$ be a Riemannian manifold which is isometrically immersed in a locally product manifold $\left(\stackrel{˜}{M},g,F\right)$. For each nonzero vector $X$ tangent to $M$ at $x$, $\theta \left(X\right)$ denotes the angle between $FX$ and ${T}_{x}M$. $M$ is said to be slant if the angle $\theta \left(X\right)$ is a constant, independent of the choice of $x\in M$ and $X\in TM$. The authors give useful characterizations of slant submanifolds in a locally product manifold.

Next the authors consider bi-slant submanifolds and, finally, semi-slant submanifolds. $M$ is called a semi-slant submanifold of $\stackrel{˜}{M}$ if there exist two orthogonal distributions ${D}_{1}$ and ${D}_{2}$ on $M$ such that $TM={D}_{1}\oplus {D}_{2}$, the distribution ${D}_{1}$ is invariant, i.e., $F\left({D}_{1}\right)={D}_{1}$ and the distribution ${D}_{2}$ is slant. The authors give necessary and sufficient condition for a submanifold $M$ of a locally product manifold $\stackrel{˜}{M}$ to be semi-slant. They also obtain integrability conditions for the distributions ${D}_{1}$ and ${D}_{2}$ mentioned above.

##### MSC:
 53B25 Local submanifolds 53C15 Differential geometric structures on manifolds