Mean Curvature of Riemannian Immersions

1. Let M and M' denote complete riemannian manifolds of dimension n and m respectively, and suppose that M is compact and oriented. For simplicity we assume that both manifolds and their metrics are smooth (i.e. of class C). In terms of local co-ordinates (x, x, ...,x") on M and local co-ordinates (y,y, •••,/") on M', the riemannian metrics are written ds = gtJ dx l dx, ds' = g'aP dy* dy * where Roman suffixes take values 1,2, ..., n and Greek suffixes take values 1,2,..., m. Let/: M ->• M' be a smooth map. Following Eells and Sampson [1], we associate with / a real number called its energy. We define an inner product on the space of 2-covariant tensors at P £ M in terms of local co-ordinates by