This question needs a bit of background: one way to study the mechanics of deformation of a continuous solid body is by defining a reference body $B_0$, a connected, well-behaved subset of $R^2$ or $R^3$. In 3D one assumes that the boundary $\partial B_0$ is a closed, orientable surface; in 2D it is a closed curve.
A motion is then a map $\psi : B_0 \rightarrow B$, where $\psi = \psi(\bar{X},t)$, $t > 0$ and $\bar{X}$ is some label (usually a position vector) for points in the reference body $B_0$. In particular, motions are treated as diffeomorphisms parametrized by $t$.
A particular subset of motions involve those that conserve volumes / areas. For instance, in a planar motion, this would mean that infinitesimal areas $dv^a \times dv^b$ are conserved by the motion, i.e. the area form is not changed by the motion. A general incompressible motion would, similarly, preserve the volume form in 3D (if I am not mistaken).
In 2D, the (area preserving) motions are symplectomorphisms, not just diffeomorphisms. One expects that this imposes very strong restrictions on the map $\psi(\bar{X},t)$. In the odd-dimension (3D), one expects similarly strong constraints from the conservation of the volume.
The governing PDE for continuous bodies is the Cauchy equation of motion
$$ \nabla \cdot \sigma + \rho \bar{b} = \rho \left ( \frac{\partial \bar{v}}{\partial t} + \bar{v} \cdot \nabla \bar{v} \right) $$
The velocity $v$ is defined as $\left. \dfrac{\partial \psi}{\partial t} \right|_X$ and $\bar{b}$ is a specified vector field. There is, in general, a highly non-trivial, local relation (a functional) that relates the stress tensor $\sigma$ and the motion.
My questions are as follows:
(1) Is the presence of an incompressibility condition associated with any non-trivial, scalar, conserved quantities in the governing equation?
(2) It is known that there are some obvious restrictions on the velocity field, for instance $\nabla \cdot \bar{v} = 0$. However, does such a condition also impose non-obvious restrictions on other geometric quantities (e.g. the associated metric tensor)
Alternatively, is there a reason that symplectomorphisms vs diffeomorphisms / conserved volume form just does not yield too much more useful information beyond the obvious?