Submanifolds

Abstract

Let M be an n-dimensional manifold isometrically immersed in an m-dimensional Riemannian manifold $\bar M$ . We put m = n+p, p > 0. We denote by $\bar \nabla$ the operator of covariant differentiation in $\bar M$ and by g the Riemannian metric tensor field in $\bar M$ . Since the discussion is local, we may assume, if we want, that M is imbedded in $\bar M$ . The submanifold M is also a Riemannian manifold with Riemannian metric h given by h(X,Y) = g(X,Y) for any vector fields X and Y on M. The Riemannian metric h on M is called the induced metric on M. Throughout this book, the induced metric h will be denoted by the same g as that of the ambient manifold $\bar M$ to simplify the notation because it may cause no confusion. Let T(M) and T(M)^⊥ denote the tangent and normal bundle of M respectively. The metric g and the connection $\bar \nabla$ on $\bar M$ lead to invariant inner products and the connections on T(M) and T(M)^⊥ We will define a connection on M explicitely.