Extrinsic curvature ($K$)

The extrinsic curvature of a hypersurface is the tangential bilinear form obtained by differentiating its unit normal. It measures how the hypersurface bends in the ambient manifold, with an overall sign depending on convention.

A hypersurface is totally geodesic when every ambient geodesic initially tangent to it remains in it. Equivalently, its extrinsic curvature vanishes identically.