a geometric form of the Radon–Nikodym theorem. No local structure A volume form on a manifold has no local structure in the sense that it is not possible on small open sets to distinguish between the given volume form and the volume form on Euclidean space . That is, for every point in there is an open neighborhood of and a diffeomorphism of onto an open set in such that the volume form on is the pullback of along As a corollary, if and are two manifolds, each with volume forms then for any points there are open neighborhoods of and of and a map such that the volume form on restricted to the neighborhood pulls back to volume form on restricted to the neighborhood : In one dimension, one can prove it thus: given a volume form on define Then the standard Lebesgue measure pulls back to under : Concretely, In higher dimensions, given any point it has a neighborhood locally homeomorphic to and one can apply the same procedure. Global structure: volume A volume form on a connected manifold has a single global invariant, namely the (overall) volume, denoted which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on On a disconnected manifold, the volume of each connected component is the invariant. In symbols, if is a homeomorphism of manifolds that pulls back to then and the manifolds have the same volume. Volume forms can also be pulled back under covering maps, in which case they multiply volume by the cardinality of the fiber (formally, by integration along the fiber). In the case of an infinite sheeted cover (such as ), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold. See also Poincaré metric provides a review of the volume form on the complex plane References . . Determinants Differential forms Differential geometry Integration on manifolds Riemannian geometry Riemannian manifolds