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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.

MSC:

53B25 Local submanifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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