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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 25th 2011

started Euler class

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 19th 2019
• (edited Mar 19th 2019)

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:

$\chi( E \oplus F ) \;=\; \chi(E) \smile \chi(F) \,.$
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 18th 2019
• (edited Apr 18th 2019)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 27th 2019
• (edited Apr 28th 2019)

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: $\big( Pf(F_\nabla) \big)^2 = det(F_\nabla)$.

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeApr 27th 2019

Why $F_\nabla$ and $F_A$?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 28th 2019

Thanks for catching this! Fixed now.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 28th 2019
• (edited Apr 28th 2019)

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 $S^{2k+1}_x \overset{\iota_x}{\hookrightarrow} S(E)$:

$\int_{S^{2k+1}} \iota_x^\ast \Omega \;=\; -1 \,.$
• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 28th 2019

finally added this classical reference (also at Pfaffian):

• Shiing-Shen Chern, A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals of Mathematics Second Series, Vol. 45, No. 4 (1944), pp. 747-752 (jstor:1969302)