= Extrinsic curvature
{title2=$K$}
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> is
$$
K(X,Y)=g(\nabla_Xn,Y)
$$
for tangent <vector fields> $X,Y$. Reversing the normal reverses $K$; some authors use the overall negative convention.
Differentiating $g(n,Y)=0$ gives $K(X,Y)=-g(n,\nabla_XY)$. 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 $\nabla_Xn=X/R$ and $K=g|_{TS}/R$. A plane has constant normal and $K=0$. The latter is the local <totally geodesic> case.
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