This paper by Jürgen Moser studies the equivalence of volume elements on a compact, connected manifold under smooth automorphisms. A volume element is a differential form of degree n that is everywhere positive. Two volume elements are equivalent if one can be transformed into the other by a smooth automorphism. The main result is that any two volume elements on a compact, connected manifold are equivalent if their total volumes are equal. This is proven by reducing the problem to a local one, where the difference between the two volume elements is small in a single coordinate patch. The proof involves constructing a diffeomorphism that transforms one volume element into another while preserving the total volume. The paper also discusses an alternative proof for closed 2-forms and notes that the problem of deforming nondegenerate 2-forms remains open. The paper concludes with a mention of how the second approach leads to a simple proof of Darboux's theorem.This paper by Jürgen Moser studies the equivalence of volume elements on a compact, connected manifold under smooth automorphisms. A volume element is a differential form of degree n that is everywhere positive. Two volume elements are equivalent if one can be transformed into the other by a smooth automorphism. The main result is that any two volume elements on a compact, connected manifold are equivalent if their total volumes are equal. This is proven by reducing the problem to a local one, where the difference between the two volume elements is small in a single coordinate patch. The proof involves constructing a diffeomorphism that transforms one volume element into another while preserving the total volume. The paper also discusses an alternative proof for closed 2-forms and notes that the problem of deforming nondegenerate 2-forms remains open. The paper concludes with a mention of how the second approach leads to a simple proof of Darboux's theorem.