# "Mathai-Quillen-type" form on $M\times M$?

Let $(M,g)$ be a compact, oriented, $(2n)$-dimensional Riemannian manifold. I'm wondering whether there is a "canonical" construction of a $(2n)$-form $\eta_g$ on $M\times M$, such that

1. $\eta_g$ is a de Rham representative of the Poincare dual of the diagonal;

2. The pullback under the diagonal inclusion $\iota:M\to M\times M$ is proportional to the curvature Pfaffian: $\iota^*\eta_g=\tfrac{1}{(2\pi)^n}\text{Pf}(\text{Rm})$.

There are plenty of "non-canonical" such forms, I think. For instance, pick a neighbourhood of the diagonal which is diffeomorphic to $TM$, and transfer the Mathai-Quillen Thom form on $TM$ to this neighbourhood using a suitable diffeomorphism. (By the way, the Mathai-Quillen Thom form on the total space of a bundle-with-metric-and-connection is the model I have in mind here for what "canonical" should mean -- a form whose value at each point depends only on local invariants.)

Motivation: Such a form would yield a proof of the Chern-Gauss-Bonnet theorem which is both quick and natural. Namely,

$\chi(M)=PD(\Delta)\cup PD(\Delta)=\int_\Delta \eta_g =\tfrac{1}{(2\pi)^n}\int_M \text{Pf}(\text{Rm})$.

I was styding the paper of Jean Michael Bismut and Christophe SOULÉ, Complex Immersions and Arakelov Geometry The following answer may help

In the case when $M$ is a vector space or co-adjoint orbit via KKS symplectic form we know the answer

Take $\pi : T^*M\to M$ be a cotangent bundle and $s:M\to T^*M$ be as section then from the work of Bismut and Soule it is known that

$$s^*\mu_0(T^*M)=(2\pi)^{-m/2}\text{Pf}(D^2)$$

Here the Thom class $\mu_t(T^*M)=(-1)^{m(m-1)/2}(2\pi)^{-m/2}B(e^{-\omega_t})$ considered when $t\to 0$ and $B\{·\}$ is the Berezin integral and $\omega_t=\frac{1}{2}t^2|x|^2+tDx+D^2$

Now in the special case when $M$ is a vector space then $T^*M\cong M\times M$ (and hence $s$ is just $\iota$). So when $M$ is the vector space this gives an answer to your question see this paper and

also

If we change your question a little bit and if we Take $M=G$ as lie gorup and consider $T^*G\cong G\times T_eG\to G$ then we can find a relation between pull-back of Kirillov–Kostant—Souriau symplectic form on coadjoint orbit $\mathcal O_\mu$ and Mathai-Quillen form. In page 11 of my Master expose in Marseille, 2014 you can find the canonical construction of KKS form and Pfaffian and by pervious construction you can relative it to Mathai-Quillen form

and also the book of Heat Kernels and Dirac Operators by Berline, Nicole, Getzler, Ezra, Vergne, Michèle http://www.springer.com/br/book/9783540200628