Non-null hypersurface projection ($P^a{}_b={\delta }^a{}_b-\epsilon n^a{n}_b,\epsilon =n_a{n}^a=\pm 1$)

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.