  We study a class of generalized Laplacian operators by violating the
ellipticity with degenerate metric tensors. The theory is motivated by the
statistical mechanics of topologically constrained particles. In the context of
diffusion models, the metric tensor is given by $g=-\mathcal{J}^2$ with a
generalized Poisson matrix $\mathcal{J}$ that dictates particle dynamics. The
standard Euclidean metric corresponds to the symplectic matrix of canonical
Hamiltonian systems. However, topological constraints bring about nullity to
$\mathcal{J}$, resulting in degeneracy in the corresponding diffusion operator;
we call such an operator an orthogonal Laplacian (since the ellipticity is
broken in the direction parallel to the nullity), and denote it by
$\Delta_{\perp}$. Although all nice properties pertinent to the ellipticity are
generally lost for $\Delta_{\perp}$, a finite helicity of $\mathcal{J}$ helps
to recover some of them by preventing foliation of space. We show that
$-\left(\Delta_{\perp}u,v\right)$ defines an inner product of a Sobolev-like
Hilbert space, and satisfies a Poincar\'e-like inequality
$-\left(\Delta_{\perp}u,u\right)\geq C\lvert\lvert{u}\rvert\rvert^{2}_{L^2}$.
Applying Riesz's representation theorem, we obtain a unique weak solution of
the orthogonal Poisson equation.
