# Topology of diffeomorphism groups which preserve the volume element.

Let $M$ be a manifold.

In the Example1.3. of Topological Methods in Hydrodynamics (V.I.Arnold, B.A. Khesin), it is written that diffeomorphisms preserving the volume element in a domain $M$ form a Lie group.

How is this topology defined? It is not explained before Example1.3..

Now I understand how to define the Whitney topology for this group $\text{SDiff}(M)$. However, are $\text{SDiff}(M)$ and $\text{Diff}$(M) differentiable manifold?
• It forms an infinite-dimensional Lie group (not a finite-dimensional one like you're probably more used to!). The topology is the subspace topology of $\text{Diff}(M)$. – user98602 Oct 12 '16 at 14:19
• What is the topology of $\text{Diff}(M)?$ – user53216 Oct 12 '16 at 14:39
You take the Whitney $C^\infty$-topology. A basis of this is given by $$S^k(U) = \{ f \in C^{\infty}(M,M) : (J^kf)(M) \subseteq U \}$$ where $k$ runs over all integers and $U$ over all open subsets of $$J^k(M,M).$$The latter is the space of $k$-jets of smooth maps $f \in C^{\infty}(M,M)$, its topology is defined in https://en.wikipedia.org/wiki/Jet_(mathematics)#Jets_of_functions_from_a_manifold_to_a_manifold