Volume preserving embeddings of open subsets of ℝⁿ into manifolds

Schlenk, Felix

doi:10.1090/S0002-9939-02-06845-4

Volume preserving embeddings of open subsets of ${\mathbb R}^n$ into manifolds
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Proc. Amer. Math. Soc. 131 (2003), 1925-1929 Request permission

Abstract:

We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $\Omega$, and we show that an open subset $U$ of ${\mathbb R}^n$ of Lebesgue measure $\operatorname {Vol}\;\! (U)$ embeds into $M$ by a smooth volume preserving embedding whenever the volume condition $\operatorname {Vol}\;\! (U) \le \operatorname {Vol}\;\!(M, \Omega )$ is met.

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