## New answers tagged differential-operators

16
votes

Accepted

### Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?

Maybe an even more elementary argument than the one of Tobias:
The continuity of all involved operators is easy: simply all differential operators with smooth coefficients between sections of vector ...

11
votes

### Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?

Yes, the Hodge decomposition is a topological decomposition with respect to the $C^\infty$-topology. One can argue, for example, that the Laplace-Beltrami $\Delta$ operator is elliptic and hence can ...

1
vote

Accepted

### Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

Consider the full bundle of exterior forms. The Hodge star operator $\ast$ acts on it with eigenvalues $\pm1$. The $-1$-eigenbundle is isomorphic to $V^{\pm1}\otimes V^-$. The map $\frac{1-\ast}2$ is ...

1
vote

### Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

It's something along the lines of spinors can be identified with the space of homomorphisms preserving some structure between two 2 dimensional complex vector spaces. If you have a sequence $0\to \...

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