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started Euler class
added statement of the Whitney sum formula for Euler classes:
The Euler class of the Whitney sum of two oriented real vector bundles to the cup product of the separate Euler classes:
$\chi( E \oplus F ) \;=\; \chi(E) \smile \chi(F) \,.$added references on Euler forms:
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)
Siye Wu, Section 2.2 of Mathai-Quillen Formalism (arXiv:hep-th/0505003)
Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)
finally added this kind of remark, to the Properties-section:
For $E$ a vector bundle of even rank $rank(E) = 2 k$, the cup product of the Euler class with itself equals the $k$th Pontryagin class
$\chi(E) \smile \chi(E) \;=\; p_k(E) \,.$(e.g. Walschap 04, p. 187)
When the Euler class is represented by the Euler form of a connection $\nabla$ on $E$, which then is fiber-wise proportional to the Pfaffian of the curvature form $F_\nabla$ of $\nabla$, the above relation corresponds to the fact that the product of a Pfaffian with itself is the determinant: $\big( Pf(F_\nabla) \big)^2 = det(F_\nabla)$.
Why $F_\nabla$ and $F_A$?
Thanks for catching this! Fixed now.
I have added statement of the following proposition, for which I am citing (Walschap 04, Chapter 6.6, Thm. 6.1, p. 201-202)
Let $X$ be a smooth manifold and $E \overset{\pi}{\longrightarrow} X$ an oriented real vector bundle of even rank, $rank(E) = 2k + 2$.
For any choice of connection $\nabla$ on $E$ ($SO(dim(X))$-connection), let $\chi(\nabla_E) \in \Omega^{2k}(X)$ denote the corresponding Euler form.
Then the pullback of the Euler form $\chi(\nabla_E)$ to the unit sphere bundle $S(E) \overset{S(\pi)}{\longrightarrow} X$ is exact
$\big( S(\pi) \big)^\ast \chi(\nabla_E) \;=\; d \Omega$such that the trivializing form has (minus) unit integral over any of the (2k+1)-sphere-fibers $S^{2k+1}_x \overset{\iota_x}{\hookrightarrow} S(E)$:
$\int_{S^{2k+1}} \iota_x^\ast \Omega \;=\; -1 \,.$finally added this classical reference (also at Pfaffian):
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