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Let $(M,g)$ be a complete, connected, oriented and noncompact Riemannian manifold. If $M$ were compact, then a vector field $X$ is called volume-preserving when its associated flow $\phi : M \times \mathbb{R} \to M$, $\phi(x,t)$, has this property, i.e., when

$$ \operatorname{vol} \phi(A, t) = \operatorname{vol} A $$

holds for all measurable $A \subset M$ and $t \in \mathbb{R}$, where $\operatorname{vol}$ denotes the volume element of $(M,g)$. How would one define a similar notion for noncompact manifolds? Do we need to suppose that the vector fields are bounded, so that they are complete? If instead we work with general vector fields, can we just impose that $A$ is bounded and measurable above?

What would be the "best"/"more appropriate" definition? Assuming this step, what can we say about the set of volume-preserving vector fields on a given complete, connected, oriented, noncompact $(M,g)$?

P.S.: I know that divergence-free vector fields are volume-preserving, at least for compact manifolds. What about for complete noncompact manifolds?

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1 Answer 1

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We can sidestep the issue about the flow not being defined uniformly across points by treating the computation infinitesimally. For any manifold (compact or now) endowed with a volume form $\omega$, the local flow of a vector field $X$ preserves $\omega$ if and only if $\mathcal L_X \omega = 0$, which again is equivalent to incompressibility (i.e., divergence-freeness) of $X$, as the divergence operator, $\operatorname{div}$, is characterized by $\mathcal L_X \omega = (\operatorname{div} X) \omega$.

Locally we can always choose coordinates $(x^1, \ldots, x^n)$ so that the volume form $\omega$ has coordinate representation $\widehat\omega = x^1 \wedge \cdots \wedge x^n$, and in those coordinates the divergence has the same form as it does for the Euclidean volume form in standard coordinates: $\operatorname{div} (F^i \partial_{x^i}) = \partial_{x^i} F^i$, where the Einstein summation convention is in effect. So, locally a volume-preserving vector field is specified by $n - 1$ functions of $n$ variables and $1$ function of $n - 1$ variables.

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