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Second Fundamental Form and Mean Curvature

  • Philippe Tondeur
Part of the Universitext book series (UTX)

Abstract

Let (M,gM) be a Riemannian manifold. For a submanifold L ⊂ M, and vector fields X,X′ tangent to L, the second fundamental form α(X,X′) takes values in the normal bundle, and is given by
$$ \alpha (X,X') = \pi (\nabla_X^MX') $$
(6.1)
where π is the projection onto the normal bundle. For a foliation F on (M,gM) this formula yields a bundle map α : L ⊗ L → Q. The involutivity of L shows that α is symmetric. In fact the definition α = - ∇π for π ∈ Ω1 (M,Q) yields even a more general symmetric form TM ⊗ TM → Q, that restricts to the α above (see [KT6, p. 94]). But here we use α in the restricted sense (6.1). Note that for Z ∈ ΓL
$$ {g_Q}(\alpha (X,X'),Z) = {g_M}(\nabla_X^MX',Z) = - {g_M}(X',\nabla_X^MZ) $$
(6.2)
From this we conclude that F is totally geodesic exactly when α = 0.

Keywords

Vector Field Riemannian Manifold Fundamental Form Characteristic Form Normal Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Philippe Tondeur
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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