Second fundamental form

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Definition

Let M be a manifold embedded in \R^n. the second fundamental form on M is the map \Gamma(TM) \times \Gamma(TM) \to \Gamma(\R^n) defined as follows:

S(X,Y) = \nabla_XY - \nabla'_XY

where \nabla si the Levi-Civita connection of \R^n and \nabla'
is the Levi-Civita connection on M.