Density bundle

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold which can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain trivial line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x.

From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates. Densities can be generalized into s-densities, whose coordinate representations become multiplied by the s-th power of the absolute value of the jacobian determinant. On an oriented manifold 1-densities can be canonically identified with the n-forms on M. On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor product of the orientation bundle of M and the n-th exterior product bundle of T*M (see pseudotensor.)

In general, there does not exist a natural concept of a "volume" for a parallelotope generated by vectors v₁, ..., v_n in a n-dimensional vector space V. However, if one wishes to define a function μ : V × ... × V → R that assigns a volume for any such parallelotope, it should satisfy the following properties:

These conditions are equivalent to the statement that μ is given by a translation-invariant measure on V, and they can be rephrased as

Any such mapping μ : V × ... × V → R is called a density on the vector space V. The set Vol(V) of all densities on V forms a one-dimensional vector space, and any n-form ω on V defines a density | ω | on V by

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