Let $(M,g)$ be a Riemannian manifold of dimension at least $4$.

We consider the differential operator $$D:\Gamma(TM)\to \Gamma (TM)$$

with $$D(X)=\nabla \circ Div(X)$$.

The principal symbol $\delta (D): \pi^* TM \to \pi^* TM$, as a bundle morphism, has constant fiber wise rank $1$ where $\pi$ is the natural projection from the unit cotangent bundle $S^*(M)$ to $M$. So the image of this bundle morphism is a line bundle over $S^*(M)$. The first Chern class of the complexification of this line bundle determines a cohomology class in $H^2(S^*M)\simeq H^2 (M)$. So we obtain a cohomology class $\lambda(M) \in H^2 (M)$.

Does this cohomology class have an alternative formulation?Is there a name for this cohomology class? In dimension $4$, does $\int_M \lambda(M) \wedge \lambda (M) $ depend on the Riemanian metric $g$ on $M$?

The motivation for this question is the following:

I was thinking to find a concept weaker than ellipticity of differential operators. Then I consider the condition that the symbol has constant rank for all fibers.(The full rank is the same as elliptic concept). The only example I found was the above differential operator.

What are some other examples of this type:i.e the symbol of our differential operator is not invertible but has constant fiber wise rank? Are there some theories which consider this generalized elliptic operators?