Geometric Structures on a Manifold and its Cotangent Bundle

The present text, Geometric Structures of a Manifold to its Cotangent Bundle, is a study of certain structures and connections in a Riemannian manifold. The study of submanifolds of an ambient manifold under the action of a (1,1)-tensor field on tangent bundle is an interesting topic in the field of differential geometry. There are two well known classes of submanifolds, namely invariant submanifolds and anti-invariant sumanifolds. Under the first case, the tangent space of the sumanifold is invariant under the action of the structural (1,1) tensor field, whereas in the second case it is mapped into the normal space. This behavior of the tangent space leads to study the geometric properties of submanifolds, invariant submanifolds and semi-invariant submanifolds of certain structures under the action of (1,1) structure tensor field. The study of horizontal and complete lifts from a manifold to its cotangent bundle is also involved.