nLab Moser's theorem

Statements

Consider a compact connected smooth manifold $X$ of dimension $d \coloneqq dim(X)$.

Recall that a volume form on $X$ is a differential form of degree $d$ which is everywhere non-degenerate.

Proposition

For $\omega, \omega' \in \Omega^d_{dR}(X)$ a pair of volume forms on $X$, there exists a diffeomorphism $\phi \colon X \longrightarrow X$ whose pullback takes one to the other: $\omega' = \phi^\ast \omega$.

Moreover, $\phi$ may be chosen to be in the connected component $Diff_0(X)$ of the identity map in the diffeomorphism group.

Related concepts

• volume preserving diffeomorphism

References

The original article:

• Jürgen Moser: On the Volume Elements on a Manifold, Trans. Amer. Math. Soc. 120 (1965) 286-294 [doi:10.1090/S0002-9947-1965-0182927-5, pdf]

Lecture notes:

• Michael Hutchings (notes by J. van Dyke): Moser’s trick (2019) [pdf]

See also:

• Wikipedia: Moser’s trick

Generalization to manifolds with corners:

• Martins Bruveris, Peter W. Michor, Adam Parusinski, Armin Rainer: Moser’s theorem on manifolds with corners, Proc. Amer. Math. Soc. 146 11 (2018) 4889-4897 [doi:10.1090/proc/14130, arXiv:1604.07787]