In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space formMm(c), c 6= 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].

In this paper, we prove that if every totally real bisectional curvature of an n(≥ 3)-dimensional complete Kähler submanifold of a complex projective space of constant holomorphic sectional curvature c is greater than c 4(n2−1)n(2n− 1), then it is totally geodesic. Mathematics Subject Classifications: 53C50, 53C55, 53C56.

We establish some inequalities for invariant submanifolds involving totally real sectional curvature and the scalar curvature in a quaternion space form. The equality cases are also discussed. AMS Mathematics Subject Classification (2010): 53C40

Involving the Ricci curvature and the squared mean curvature, we obtain basic inequalities for different kind of submaniforlds of a Sasakian space form tangent to the structure vector field of the ambient manifold. Contrary to already known results, we find a different necessary and sufficient condition for the equality for Ricci curvature of C-totally real submanifolds of a Sasakian space form...

The index of a metric plays significant roles in differential geometry as it generates variety of vector fields such as space-like, time-like, and light-like fileds. With the help of these vector fields, we establish interesting properties on ( )-Sasakian manifolds, which was introduced by Bejancu and Duggal [1] and further investigated by Xufeng and Xiaoli [2]. Since Sasakian manifolds with in...