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Let M {\displaystyle M} be a manifold embedded in R n {\displaystyle \mathbb {R} ^{n}} . the second fundamental form on M {\displaystyle M} is the map Γ ( T M ) × Γ ( T M ) → Γ ( R n ) {\displaystyle \Gamma (TM)\times \Gamma (TM)\to \Gamma (\mathbb {R} ^{n})} defined as follows:
S ( X , Y ) = ∇ X Y − ∇ X ′ Y {\displaystyle S(X,Y)=\nabla _{X}Y-\nabla '_{X}Y}
where ∇ {\displaystyle \nabla } si the Levi-Civita connection of R n {\displaystyle \mathbb {R} ^{n}} and ∇ ′ {\displaystyle \nabla '} is the Levi-Civita connection on M {\displaystyle M} .
