Tensor density is a concept from the field of differential geometry and tensor analysis, and it arises in the context of general relativity and manifold theory. Tensors are mathematical objects that can be used to represent physical quantities, and they can be defined at each point of a manifold, which is a space that is locally similar to Euclidean space.
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A tensor density of weight transforms like a tensor with an additional factor under . A scalar density of weight one can be integrated against the coordinate volume element to give an invariant integral. On a spatial hypersurface, is such a scalar density; hence is a tensor density when is an ordinary tensor. The Levi-Civita connection extends to densities and satisfies .