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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 M ¯ is said to be slant if for each non zero vector XT p M the angle θ(X) between JX and T p M is independent of the choice of pM and XT p M. Consider a Kähler product manifold M ¯=M ¯ m ×M ¯ n . Denote by P ¯ and Q ¯ the projection operators of the tangent space of M ¯ to the tangent spaces of M ¯ m and M ¯ n , respectively, and put F=P ¯-Q ¯. It is proved that an F-invariant, slant submanifold of a Kähler product manifold M ¯=M ¯ m ×M ¯ n is a product manifold M 1 ×M 2 and M 1 (resp., M 2 ) is also a slant submanifold of M ¯ m (resp., M ¯ n ). It is also obtained that if M 1 ×M 2 is the Kähler slant submanifold of M ¯ then M 1 (resp., M 2 ) is a Kähler slant submanifold of M ¯ m (resp., 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 M ¯ m (c 1 )×M ¯ n (c 2 ) are obtained.
MSC:
53C15Differential geometric structures on manifolds
53C40Global submanifolds (differential geometry)