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

Manifold 

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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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