Abstract
Suppose f: M → ℝn is a smooth immersion of an m-dimensional manifold M in ℝn, 1 ≤ m ≤ n. If (p 1,...,p m) are local coordinates in M, then the vectors f i = ∂f/∂p i constitute a basis in the tangent m-dimensional plane T to (M, f) at the point p. When n > m we can choose an orthonormed basis (v 1 ,..., v n−m ) in the (n − m)-dimensional orthogonal complement N to T. Each unit normal v ∈ N can be assigned the second fundamental form of the m-dimensional surface (M, f) by using the equality II (v) = <d 2 f, v>. Denote mH j = trII(v j ) = k 1 j +···k mj , where tr is the trace of the form; k1 j ,···,k mj are the principal curvatures of (M, f) with respect to the normal v j , i.e. the eigenvalues of II(v j ). The vector
(the sum being taken over j from 1 to n − m) does not depend on the choice of orthonormed basis {v j } in N. This vector H is said to be the mean curvature vector of the m-dimensional surface (M, f) at the point p ∈ M and its norm
is the absolute mean curvature.
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© 1988 Springer-Verlag Berlin Heidelberg
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Burago, Y.D., Zalgaller, V.A. (1988). Immersions in ℝn . In: Geometric Inequalities. Grundlehren der mathematischen Wissenschaften, vol 285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07441-1_5
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DOI: https://doi.org/10.1007/978-3-662-07441-1_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05724-3
Online ISBN: 978-3-662-07441-1
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