nForum - Discussion Feed (Euler class) 2023-09-28T10:20:20+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "Euler class" (77653) https://nforum.ncatlab.org/discussion/3047/?Focus=77653#Comment_77653 2019-04-28T16:34:35+00:00 2023-09-28T10:20:19+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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)
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Urs comments on "Euler class" (77639) https://nforum.ncatlab.org/discussion/3047/?Focus=77639#Comment_77639 2019-04-28T11:10:52+00:00 2023-09-28T10:20:19+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 XX be a smooth manifold and E&longrightarrow;&pi;XE ...

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

$\int_{S^{2k+1}} \iota_x^\ast \Omega \;=\; -1 \,.$ ]]>
Urs comments on "Euler class" (77634) https://nforum.ncatlab.org/discussion/3047/?Focus=77634#Comment_77634 2019-04-28T09:12:55+00:00 2023-09-28T10:20:19+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks for catching this! Fixed now.

Thanks for catching this! Fixed now.

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DavidRoberts comments on "Euler class" (77623) https://nforum.ncatlab.org/discussion/3047/?Focus=77623#Comment_77623 2019-04-27T21:58:53+00:00 2023-09-28T10:20:19+00:00 DavidRoberts https://nforum.ncatlab.org/account/42/ Why F &Del;F_\nabla and F AF_A?

Why $F_\nabla$ and $F_A$?

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Urs comments on "Euler class" (77619) https://nforum.ncatlab.org/discussion/3047/?Focus=77619#Comment_77619 2019-04-27T17:55:56+00:00 2023-09-28T10:20:19+00:00 Urs https://nforum.ncatlab.org/account/4/ finally added this kind of remark, to the Properties-section: For EE a vector bundle of even rank rank(E)=2krank(E) = 2 k, the cup product of the Euler class with itself equals the kkth ...

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

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Urs comments on "Euler class" (77445) https://nforum.ncatlab.org/discussion/3047/?Focus=77445#Comment_77445 2019-04-18T10:23:00+00:00 2023-09-28T10:20:20+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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Urs comments on "Euler class" (76609) https://nforum.ncatlab.org/discussion/3047/?Focus=76609#Comment_76609 2019-03-19T11:54:17+00:00 2023-09-28T10:20:20+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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:

$\chi( E \oplus F ) \;=\; \chi(E) \smile \chi(F) \,.$ ]]>
Urs comments on "Euler class" (25436) https://nforum.ncatlab.org/discussion/3047/?Focus=25436#Comment_25436 2011-08-25T17:31:15+00:00 2023-09-28T10:20:20+00:00 Urs https://nforum.ncatlab.org/account/4/ started Euler class

started Euler class

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