Shape Representation as the Intersection of n-k Hypersurfaces

We investigate the feasibility of representing implicitly a k-dimensional manifold embedded in the Euclidean space $\mathbb{R}^n$ as the intersection of n-k transverse hypersurfaces. From the analytical point of view, the embedded manifold is defined as the inverse image of a regular value of a vector function. This approach is a priori appealing since the corresponding function is differentiable at any point of the embedded manifold. We focus on time-dependent manifolds and establish the link between the velocity field of the evolving manifold and a Partial Differential Equation (PDE) satisfied by its describing function.