Sorry. I was stupid to miss this condition.

What's the definition of the anti-canonical spin structure?

1. If you mean the DEGREE of the line bundle, i.e., the first Chern class. Then that of the canonical bundle is equal to -(N+1).

You may refer to the book Lectures on Chern-Weil theory and Witten deformation section 1.8.

@ stankewicz Thank you. It seems that the Borel-Moore homology is kind of homology with compact support. But what kind of duality is it? Can you provide some reference?

@ Tom Goodwillie What do you mean more than one thing? Yes, for $M\times_G EG$ is not good in that sense.

I know nothing about Finsler geometry. And other than Mathai-Quillen's Thom form, I don't know other "geometrically meaningful" forms with your condition.

How about you restate your question? It's not that convenient to discuss like this.

I think the Mathai-Quillen case is that $V=TM$ and its a special Thom form(exponential decay in vertical direction). For Gauss-Bonnet, you don't have a section of the sphere bundle in general but a section with singularities.

Sorry I didn't make it clear. If $V$ has nothing to do with $TM$, then if $e$ is an Euler form of $TM$, then $p^*e$ is a differential form on $V$, and then for any section $s$ of $V$, obviously $s^* p^* e= (ps)^* e=e$, by tautology. But this is not much useful. In case $V=TM$ or the sphere bundle of $TM$, we have Thom forms or angular forms respectively. The pull-back of the former gives an Euler form since it can be viewed as a definition of Euler class. For the latter(sphere bundle), if you have a section, then the Euler characteristic is zero.