We consider a Riemannian manifold $(M,g)$ and the metric can induce the dual $d^*$ of $d$ by the formula: $$ \int_M \langle\alpha, d\beta\rangle d{\text vol}_g = \int_M \langle d^*\alpha, \beta\rangle d{\text vol}_g $$
I am a new graduate student, and one of my teacher mentioned in a class that $$d+d^*$$ is an elliptic operator. I am curious about this. I know elliptic operators in Euclidean spaces, but how to understand this? Could you tell me some references about this issue? Thanks.
Note that on an Euclidean space, one is usually more familiar with the notion of a scalar elliptic operator (like the regular Laplacian which acts on functions) while in your situation, the operator is described in local coordinates as a vector operator and the solutions of $\Delta_g \omega = 0$ or $(d + d^{*})(\omega) = 0$ are described in local coordinates as solutions of a system of PDEs. Hence, one needs to generalize the (maybe) familiar notion of ellipticity to the case of a system of PDEs and then one can interpret your statement accordingly.
To do this invariantly on a manifold, one needs the notion of a partial differential operator between sections of vector bundles (which, in local coordinates, amounts to a system of partial differential operators) and to define when such an operator is elliptic. A good reference for the topic is Chapter 10 of Liviu I. Nicolaescu's Lectures on the Geometry of Manifolds.