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.
KeywordsVector Field Riemannian Manifold Sectional Curvature Fundamental Form Normal Bundle
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