Hypersurface

A hypersurface in an $n$-dimensional smooth manifold is a smooth submanifold of dimension $n-1$. Locally, a regular level set $\{\varphi =0\}$ with $\nabla \varphi \ne 0$ is a hypersurface whose normal vector is $\nabla \varphi$.

A non-null hypersurface projection annihilates the unit normal vector and fixes tangent vectors to the hypersurface. It obeys $P^2=P$. Apply it to every tensor index to remove normal components, with the metric used to lower or raise the index pair for covariant slots. For a timelike unit normal in mostly-plus signature it is the spatial projection tensor. A null normal cannot be normalized this way, and a null hypersurface requires additional choices for a corresponding projection.

Local defining functions $f_{\alpha }$ for a properly embedded hypersurface $Y\subset X$ have nowhere-zero smooth ratios $f_{\alpha }/f_{\beta }$ on overlaps. These ratios define a real line bundle with a global transverse section whose local representatives are the $f_{\alpha }$ and whose zero set is $Y$. The bundle is naturally isomorphic to the normal bundle of $Y$.

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.