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Semi-slant submanifolds of a locally product manifold. (English) Zbl 1093.53025
Let $(\widetilde M,g,F)$ 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: $$F^2=I\,,\ \ g(FX,FY)=g(X,Y)\,,\ \ \widetilde\nabla F=0\,,\text{ for } X,Y\in T\widetilde M\,,$$ $\widetilde\nabla$ being the Levi-Civita connection on $\widetilde M$. 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 $(\widetilde M,g,F)$. For each nonzero vector $X$ tangent to $M$ at $x$, $\theta(X)$ denotes the angle between $FX$ and $T_xM$. $M$ is said to be slant if the angle $\theta(X)$ 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 $\widetilde M$ 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., $F(D_1)=D_1$ and the distribution $D_2$ is slant. The authors give necessary and sufficient condition for a submanifold $M$ of a locally product manifold $\widetilde M$ to be semi-slant. They also obtain integrability conditions for the distributions $D_1$ and $D_2$ mentioned above.

53B25Local submanifolds
53C15Differential geometric structures on manifolds