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On an oriented $d$-dimensional pseudo-Riemannian manifold, the metric volume tensor is the totally antisymmetric tensor with ${ϵ}_{1\dots d}=\sqrt{|\det g|}$ in positively oriented coordinates. It supplies the components of the volume form. It differs from the coordinate Levi-Civita symbol, whose entries are just $0,1,-1$. Raising indices with a Lorentzian metric can change signs; the metric, dimension and orientation must be specified.

A pseudo-Riemannian manifold is a smooth manifold with a smooth metric tensor that is a nondegenerate bilinear form on each tangent space. Positivity is not required. A Riemannian manifold is the positive-definite case, while a Lorentzian manifold has exactly one temporal direction in its metric signature.