Geometry of indefinite manifolds and their related structures

Abstract

The theory of manifolds is an old branch of differential geometry. Beginning with the theory of curves and surfaces and the present research entitled “Geometry of indefinite manifolds and their related structures” deals with the theory of manifolds and their induced structures when they are endowed with indefinite metrics. During the last few decades the study of differentiable manifolds with indefinite metric attracted a society of mathematicians because of its applications in General Relativity and Relativistic Physics. Also because of the signature of the metric we expect essential changes in the study of differentiable manifolds. Hence the study of differentiable manifolds with indefinite metrics becomes the central theme in present scenario. Since the study of curvature tensor with indefinite metrics plays a fundamental role in physics hence it becomes necessary to study manifolds endowed with indefinite metric. In our research, we shall focus our attention towards this direction and will study the equivalence classes of holomorphic sectional curvature, antiholomorphic sectional curvature and bisectional curvature for various types of indefinite manifolds. Since Sasakian manifolds with indefinite metrics play crucial roles in physics, hence we shall study the geometry of Sasakian manifolds and their structures with indefinite metrics. As the general expression for Riemannian curvature tensor for different classes of almost Hermitian manifolds with constant holomorphic sectional curvature in terms of metric tensor is not yet known.