# “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,

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

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