 Extrinsic curvature

For a non-null hypersurface in a pseudo-Riemannian manifold, choose a unit normal vector $n$ and the ambient Levi-Civita connection $\nabla$. One sign convention for its extrinsic curvature isfor tangent vector fields $X,Y$. Reversing the normal reverses $K$; some authors use the overall negative convention.

Differentiating $g(n,Y)=0$ gives $K(X,Y)=-g(n,{\nabla }_{X}Y)$. The torsion-free property and tangency of $[X,Y]$ show that $K$ is a symmetric bilinear form. It measures the normal part of ambient acceleration along the hypersurface.

For a sphere of radius $R$ in Euclidean SPACE, the outward unit normal is $n(x)=x/R$, so ${\nabla }_{X}n=X/R$ and $K=g{|}_{TS}/R$. A plane has constant normal and $K=0$. The latter is the local totally geodesic case.