## Heat kernel proof of Poincaré Hopf

I am familiar with Witten's proof of the Morse inequalities by semiclassical analysis. Hey uses semiclassical expansions of the first eigenfunctions to construct the Morse complex from it, which implies Poincaré-Hopf.

Is there any proof of the theorem that uses a similar approach as in the atiyah-singer index theorem, by replacing the usual heat kernel by a parameter dependent one?

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 Yes, there exist such a proof. You can find it in Bismut's paper "The Witten complex and the degenerate Morse inequalities." projecteuclid.org/… But I would like to say he also use the Morse complex but more closed to the heat kernel method. The essential is the same as Witten's proof... – shu Dec 13 at 11:46