Second fundamental form

Definition

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

$S(X,Y)={\mathrm{\nabla }}_{X}Y-\mathrm{\nabla }{\text{'}}_{X}Y$

where $\mathrm{\nabla }$ si the Levi-Civita connection of ${\mathbb{R}}^n$ and ${\mathrm{\nabla }}^{\prime }$ is the Levi-Civita connection on $M$.