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Mean Curvature of Riemannian Immersions
Author(s) -
Willmore T. J.
Publication year - 1971
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-3.2.307
Subject(s) - willmore energy , citation , curvature , mathematics , scalar curvature , sectional curvature , computer science , world wide web , geometry
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