When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ (adjoint takes into account the metric). This operator is self adjoint and interchanges odd and even degree forms: therefore is of antidiagonal form. The index of $(d+d^*)^+$ is equal to the Euler characteristic of $M$: $$ind(d+d^*)^+=\chi(M)$$

- Can we derive this formula from the right hand side of the Atiyah Singer index formula (involving Todd genus and Chern character)?

Moreover there is so called *generalized* Gauss Bonnet theorem which relates Euler characteristic to the geometric quantity $\int_{M}Pf(\Omega)$ (Pfaffian of the curvature form): $$\chi(M)=(2\pi)^{-n}\int_MPf(\Omega).$$
So we have at all *three* quantities: the index, Euler characteristic and the integral of Pfaffian.

- Is it possible to obtain the equality $$ind(d+d^*)^+=(2\pi)^{-n}\int_MPf(\Omega)$$
withoutinvoking Gauss-Bonnet (which as far as I know requires some work).

The first question is something like "first step" in order to get into index theory and to get some better understanding of the background of Atiyah Singer index theorem. Second question is inspired by the following: $\chi(M)$ is a topological quantity: from the other hand $Pf(\Omega)$ is of geometric flavour. Gauss-Bonnet formula relates these two quantities: but as $d^*$ requires choice of the metric, $ind(d+d^*)^+$ should also be viewed as *geometric* (or maybe better, *analytical* quantity) so maybe it would be easier to relate this two *geometric* quantities and *then* refer to the Euler characteristic.