In the context of incompressible elasticity, I failed at finding a Lie group structure, which respects the incopressibility.
Question:
Let $\mathcal B, \mathcal S$ be Riemannian manifolds and $\overline{\mathcal B}$ is compact.
Does the smooth manifold of incompressible deformations $$ \mathcal{C} := \{ \varphi: \mathcal{B} \to \mathcal S \mid \varphi \in \mathrm{Diff}(\mathcal B, \mathcal S) ~\text{with}~ \mathrm{det}(\mathrm{D} \varphi) = 1\} $$ admit a Lie group structure? (Here, $\mathcal B$ is the reference configuration of the body (fixed in time) and $\mathcal S$ is the space in which the body is placed, i.e. $\varphi(X)$ is the current position of the material point $X$.)
Is a Lie structure only possible, if for example $\mathcal S \subseteq \mathbb{R}^d$ or if $\mathcal S$ is a Lie group?
Recommendations for related literature or a short 'No, there is no Lie group' are very welcome!
My problem:
A reasonable candidate for the Lie group operation is the composition of maps. But if $\mathcal B \neq \mathcal S$, it is tricky to define $\varphi_2 \circ \varphi_1$ for $\varphi_i \in \mathrm{Diff}(\mathcal B, \mathcal S)$.
Fluid dynamics:
In fluid dynamics, there seem to exists a Lie group, since typically the Eulerian perspective is used. If $\Omega$ denotes the domain of the fluid, then the displacement of fluid particles is a map $\phi \in \mathrm{Diff}(\Omega, \Omega)$, which is a Lie group with the composition of two maps as group operation.
But I was not able to transfer this to the setting above.