Pseudo-Riemannian metric ($g\in \Gamma ({\mathrm{Sym}}^2{T}^{\ast }M)$)

A pseudo-Riemannian metric is a smooth nondegenerate symmetric bilinear form on every tangent space, with locally constant signature. Its positive-definite special case is a Riemannian metric. Unlike a positive-definite metric, an indefinite metric admits nonzero null vectors. In either signature its Levi-Civita connection is uniquely determined by torsion-freeness and metric compatibility.