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 $\lambda(M) \in H^2 (M)$.



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> 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 $M$?
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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.



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> 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?
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A second cohomology class associated to a Riemannian manifold
Up to sign, the principal symbol of the operator in question is the projection onto the tautological line bundle in $\pi^\*TM$. To see this, identify $TM$ with $T^\*M$ using the Riemannian metric, and compute the symbol at $w\in S(TM)$. The symbol of the divergence at $w$ is $\langle-,w\rangle$. The symbol of the gradient is $w$ itself. The composition is the projection onto the trivial real line bundle $L\subset\pi^\*TM$ over $S(TM)$ that is spanned by $w$ at $w\in S(TM)$. All characteristic classes of $L$ vanish.


A first order with constant rank symbol, extending your example, would be the total differential $d\colon\Omega^\bullet(M)\to\Omega^\bullet(M)$. If $M$ is a surface, it is not hard to see that the symbol class becomes trivial over $S(TM)$. 


