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

Let (M ˜,g,F) be a C 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), ˜F=0,forX,YTM ˜,

˜ being the Levi-Civita connection on 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 (M ˜,g,F). For each nonzero vector X tangent to M at x, θ(X) denotes the angle between FX and T x M. M is said to be slant if the angle θ(X) is a constant, independent of the choice of xM and XTM. 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 M ˜ if there exist two orthogonal distributions D 1 and D 2 on M such that TM=D 1 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 M ˜ to be semi-slant. They also obtain integrability conditions for the distributions D 1 and D 2 mentioned above.

MSC:
53B25Local submanifolds
53C15Differential geometric structures on manifolds