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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 25th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarted [[Euler class]]

   started Euler class

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 19th 2019
   • (edited Mar 19th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement of the Whitney sum formula for Euler classes: *** The Euler class of the [[Whitney sum]] of two [[orthogonal group|oriented]] [[real vector bundles]] to the [[cup product]] of the separate Euler classes: $$ \chi( E \oplus F ) \;=\; \chi(E) \smile \chi(F) \,. $$ <a href="https://ncatlab.org/nlab/revision/diff/Euler+class/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Euler+class/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Euler+class">current</a>

   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:

   $$\chi( E \oplus F ) \;=\; \chi(E) \smile \chi(F) \,.$$

   diff, v5, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 18th 2019
   • (edited Apr 18th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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 (<a href="https://doi.org/10.1016/0040-9383(86)90007-8">10.1016/0040-9383(86)90007-8</a>) * Siye Wu, Section 2.2 of _Mathai-Quillen Formalism_ ([arXiv:hep-th/0505003](https://arxiv.org/abs/hep-th/0505003)) * [[Liviu Nicolaescu]], Section 8.3.2 of _Lectures on the Geometry of Manifolds_, 2018 ([pdf](https://www3.nd.edu/~lnicolae/Lectures.pdf), [MO comment](https://mathoverflow.net/a/117982/381)) <a href="https://ncatlab.org/nlab/revision/diff/Euler+class/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/Euler+class/10">v10</a>, <a href="https://ncatlab.org/nlab/show/Euler+class">current</a>

   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)

   diff, v10, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2019
   • (edited Apr 28th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexfinally added this kind of remark, to the Properties-section: *** For $E$ a [[vector bundle]] of [[even number|even]] [[rank of a vector bundle|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](https://ncatlab.org/nlab/show/Euler+class#Walschap04)) 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]]: $\big( Pf(F_\nabla) \big)^2 = det(F_\nabla)$. <a href="https://ncatlab.org/nlab/revision/diff/Euler+class/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Euler+class/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Euler+class">current</a>

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

   diff, v12, current

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 27th 2019
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexWhy $F_\nabla$ and $F_A$?

   Why $F_\nabla$ and $F_A$?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 28th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for catching this! Fixed now.

   Thanks for catching this! Fixed now.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 28th 2019
   • (edited Apr 28th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added statement of the following proposition, for which I am citing ([Walschap 04, Chapter 6.6, Thm. 6.1, p. 201-202](https://ncatlab.org/nlab/show/Euler+class#Walschap04)) *** Let $X$ be a [[smooth manifold]] and $E \overset{\pi}{\longrightarrow} X$ an oriented [[real vector bundle]] of [[even number|even]] [[rank of a vector bundle|rank]], $rank(E) = 2k + 2$. For any choice of [[connection on a bundle|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 differential forms|pullback]] of the [[Euler form]] $\chi(\nabla_E)$ to the [[unit sphere bundle]] $S(E) \overset{S(\pi)}{\longrightarrow} X$ is [[exact differential form|exact]] $$ \big( S(\pi) \big)^\ast \chi(\nabla_E) \;=\; d \Omega $$ such that the trivializing form has (minus) unit [[integration of differential forms|integral]] over any of the [[n-sphere|(2k+1)-sphere]]-[[fibers]] $S^{2k+1}_x \overset{\iota_x}{\hookrightarrow} S(E)$: $$ \int_{S^{2k+1}} \iota_x^\ast \Omega \;=\; -1 \,. $$ <a href="https://ncatlab.org/nlab/revision/diff/Euler+class/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Euler+class/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Euler+class">current</a>

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

   $$\int_{S^{2k+1}} \iota_x^\ast \Omega \;=\; -1 \,.$$

   diff, v15, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 28th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexfinally 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](https://www.jstor.org/stable/1969302)) <a href="https://ncatlab.org/nlab/revision/diff/Euler+class/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/Euler+class/16">v16</a>, <a href="https://ncatlab.org/nlab/show/Euler+class">current</a>

   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)

   diff, v16, current