Let $V$ is a vector bundle or sphere bundle over smooth manifolds $M$. We can get an pullback bundle over $V$ by $p^*TM$ , $p: V \rightarrow M$ . If we give the inner product on each fiber of $p^*TM$ , is there a differetial form on $V$ such that it pullback by a section of $V$ is the Euler form ? I know two famous examples, frist is Mathai-Quillen Thom form, second is Causs-Bonnet-Chern type formula on Finsler manifolds. Is there some other examples?

Edit: May be my expression is not exactly. I restate it now. The first, I express why I have this question. If $M$ is a Riemannian manifold, we can use the connection of $TM$ to construct a differetial form that is Pfaffian, it's integration on $M$ is Euler characteristic. This is Gauss-Bonnet-Chern formula. If $M$ is a Finsler manifold, we must use pullback bundle $p^*TM$ on sphere bundle, then use the preserves connection of $p^*TM$ can get a Gauss-Bonnet-Chern type formula, that is a differential form on sphere bundle and pullback by a section with isolated singular can get a "Euler form"(I mean a differetial form, it's integration on $M$ is Euler characteristic) on $M$. About this you can read "A Gauss-Bonnet-Chern formula for Finsler manifolds" by Z.Shen http://www.math.iupui.edu/~zshen/Research/preprintindex.html We also can use torsion-free connection of $p^*TM$ to get a Gauss-Bonnet-Chern type formula, you can read On the Gaussâ€“Bonnet Formula in Riemannâ€“Finsler Geometry by B.Lackey. If we think Mathai-Quillen Thom form, we use pullback bundle $p^*TM$ on $TM$, and the preserves connection of $p^*TM$ can use to construct a differential form on $TM$ we need. I think the metric on $p^*TM$ is the thing we must use, and we needn't restrict on $TM$ or $SM$, we can use another vector bundle or sphere bundle to get a pullback bundle on it.

The second, I turn to my question. If $V$ is a vector bundle or sphere bundle over smooth manifolds $M$. $p^*TM$ is a pullback bundle on $V$, I want to use the connection of this pullback bundle to construct a differential form on $V$ and use a section to pullback it, then get a "Euler form".