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cross-linked with Euler form and added these pointers:
Discussion of Euler forms (differential form-representatives of Euler classes in de Rham cohomology) as Pfaffians of curvature forms:
{#MathaiQuillen86} Varghese Mathai, Daniel Quillen, below (7.3) of Superconnections, Thom classes, and equivariant differential forms, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)
{#Wu05} Siye Wu, Section 2.2 of Mathai-Quillen Formalism, pages 390-399 in Encyclopedia of Mathematical Physics 2006 (arXiv:hep-th/0505003)
Hiro Lee Tanaka, Pfaffians and the Euler class, 2014 (pdf)
{#Nicolaescu18} Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)
added the actual statement that $Pf^2 = det$ (here) and a pointer to explicit proofs in
I gather a proof is also spelled out somewhere in Spivak’s Comprehensive Introduction to Differential Geometry but I haven’t tracked down the section/page number yet.
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