Immersions into special types of Riemannian manifolds

Abstract

Levi-Civita has proved that given a manifold M with a Rieman­ nian metric g there is a unique connection 'V satisfying the follow­ ing two properties: • 'V g = 0, i.e., the connection is metric.

• For X, Y E x(M), T(X, Y) = 0, i.e., the connection is sym- metric. The connection is then called a Riemannian connection.

Friedmann (1924) and Schouten (1954) [19] introduced the idea of semi-symmetric linear connection on a differentiable manifold. Hayden (1932) [11] introduced semi-symmetric metric connection on a Riemannian manifold and this was further developed by Yano (1970) [20], Imaii (1972) [13], Nakao (1976) [16], Amur and Pujar (1978) [4]. ln 1992, Agashe and Chafle [2] defined a semi-symmetric non-

metric connection 'V*

on a Riemannian manifold M and defined the

curvature tensor of M with respect to this semi-symmetric non- metric connection. They obtained a relation connecting the curva­ ture tensors of M with respect to semi-symmetric non-metric con­ nection and the Riemannian connection.