Pseudo-Riemannian metric
= Pseudo-Riemannian metric
{title2=$g\in\Gamma(\operatorname{Sym}^2T^*M)$}
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.