Let $G$ be a Lie group and $\pi:E_G\rightarrow M $ be a principal $G$-bundle.

I have seen in many places that a connection on $(E_G,M,G)$ is a splitting of the Atiyah sequence

$$ 0\rightarrow \text{ad}(E_G)\rightarrow \text{At}(E_G)\rightarrow T M\rightarrow 0$$

where $\text{ad}(E_G)$ is the adjoint vector bundle for $E_G$ and $\text{At}(E_G)$ is the Atiyah bundle for $E_G$.

Reference given for this is Atiyah's paper. This paper is slightly difficult to read.

Can some one give an outline of this construction or point out some exposition where this is written in detail.

  • 2
    $\begingroup$ Chen-Stiénon-Xu (2016) is quite good and further recommends Kapranov (1999). Both are on arXiv. There is also Biswas-Raghavendra (2008; pdf). $\endgroup$ – Francois Ziegler May 5 '19 at 17:58
  • $\begingroup$ @FrancoisZiegler sir, thank you for references. I will see and tell which I found it useful for my purpose... thank you... $\endgroup$ – Praphulla Koushik May 6 '19 at 1:36
  • $\begingroup$ Can some one tell me what is the reason for downvote? $\endgroup$ – Praphulla Koushik May 7 '19 at 4:18

First note that the adjoint bundle $ad(E_G)$ can be canonically identified with the vertical tangent bundle $V E_G / G$: send the pair $(p, \xi)$ consisting of a point $p \in E_G$ and a Lie algebra element $\xi$ to the value $p \cdot \xi$ at $p$ of the fundamental vector field generated by $\xi$. Moreover, the Atiyah bundle $At(E_G)$ is just a fancy way of writing $T E_G / G$.

Thus, a splitting of the Atiyah sequence is nothing else than a diffeomorphism of $T E_G / G$ with the direct sum $V E_G /G \oplus T M$ (taken over $M$). In this way, you recover the definition of a connection as a complement to $V E_G$. The horizontal bundle is the image of $TM$ under the above isomorphism $V E_G \oplus T M \to T E_G / G$. Alternatively, you can view a splitting as a projection onto $V E_G$ or as a $G$-equivariant lift $TM \to TE_G$. These equivalent viewpoints give you the definition of a connection as a connection $1$-form and as a horizontal lift operator, respectively. I've expanded a bit on the different but equivalent viewpoints in an answer to a different question.

  • $\begingroup$ Hi, thanks for your answer...if $ad(E_G)$ is identified with vertical tangent bundle $VE_G$ and Atiyah bundle $At(E_G)$ is fancy way of writing the quotient $TE_G/G$ then, I think I understand what this Atiyah sequence has to do with connection.. I will try to write down how “$At(E_G)$ is fancy way of writing the quotient $TE_G/G$” thanks thanks... $\endgroup$ – Praphulla Koushik May 7 '19 at 4:40
  • $\begingroup$ Can you give some reference to read about this Atiyah sequence in more detail? $\endgroup$ – Praphulla Koushik May 10 '19 at 17:41
  • $\begingroup$ You mean $\frac{TE_G}{G}\cong VE_G\oplus TM$ and not $TE_G \cong VE_G\oplus TM$. right? Am I missing something? $\endgroup$ – Praphulla Koushik May 12 '19 at 10:03
  • $\begingroup$ Yes, thanks for pointing this out. Changed my answer accordingly. $\endgroup$ – Tobias Diez May 12 '19 at 21:53
  • $\begingroup$ Oh.. Ok.. Thanks :) $\endgroup$ – Praphulla Koushik May 13 '19 at 5:19

See the paper The Atiyah bundle and connections on a principal bundle by Indranil Biswas.

Let $p:E_G\rightarrow M$ be a principal $G$-bundle. Michael Atiyah in his paper uses a exact sequence of vector bundles over $M$, namely $$0\rightarrow \text{ad}(E_G)\rightarrow \text{At}(E_G)\rightarrow TM\rightarrow 0$$ to define a connection on the principal bundle as a section of above exact sequence. Here,

  • $\text{ad}(E_G):(E_G\times\mathfrak{g})/G\rightarrow M$ is the adjoint bundle associated to the principal $G$-bundle $E_G\rightarrow M$ with adjoint action of $G$ on $P$.
  • $\text{At}(E_G):TE_G/G\rightarrow M$ is the Atiyah bundle. Consider the tangent bundle $TE_G\rightarrow E_G$. Action of $G$ on $E_G$ induce an acion of $G$ on $TE_G$ as well. This map $TE_G\rightarrow E_G$ is $G$-equivariant, giving a vector bundle $TE_G/G\rightarrow E_G/G$. As $E_G/G\cong M$, this gives vector bundle $TE_G/G\rightarrow M$, a vector bundle over $M$.

The map $E_G\times \mathfrak{g}\rightarrow TE_G$ given by $(p,A)\rightarrow (\delta_p)_{*,e}(A)$ where $\delta_p:G\rightarrow E_G$ is given by $g\mapsto pg$. Even this map $E_G\times \mathfrak{g}\rightarrow TE_G$ is $G$-equivariant, giving $(E_G\times \mathfrak{g})/G\rightarrow TE_G/G$. This map of manifolds is good enough to give a morphsim $\text{ad}(E_G)\rightarrow \text{At}(E_G)$ of vector budnles over $M$. The map $P\rightarrow M$ gives $TP\rightarrow TM$. This map is $G$-invarinat, giving $TP/G\rightarrow TM$. Again, this map is good enough to give a morphism $\text{At}(P)\rightarrow TM$ of vector bundles over $M$.

Combining above morphisms of vector bundles over $M$, this gives an exact sequence of vector bundles $$0\rightarrow \text{ad}(P)\rightarrow \text{At}(P)\rightarrow TM\rightarrow 0.$$

Then, author says, a connection is splitting of above exact sequence.

There are two ways (that I know) to think about splitting. One way is to talk about the direct sum $\text{At}(E_G)\cong \text{ad}(E_G)\oplus TM$ and the second is a section $s:TM\rightarrow \text{At}(E_G)$.

To relate to the notion of connection that we already know (as a distribution $\mathcal{H}\subset TE_G$) satisfying some properties, it is useful to think of splitting as a section $s:TM\rightarrow \text{At}(E_G)$. The map $p:E_G\rightarrow M$ pullback the bundle $\text{At}(E_G)\rightarrow M$ to give $p^*(\text{At}(E_G))\rightarrow E_G$ and pullback the bundle $TM\rightarrow M$ to give $p^*TM\rightarrow E_G$. Further, the map $s:TM\rightarrow \text{At}(E_G)$ is pullback to give $D:p^*TM\rightarrow p^*(\text{At}(E_G))$.

In page number $301$ (equation $2.7$), they defines a morphism of vector bundles $\mu:p^*(\text{At}(E_G))\rightarrow TE_G $ and observes that this is an isomorphism of vector bundles.

Consider the composition $p^*TM\xrightarrow{D} p^*(\text{At}(E_G))\xrightarrow{\mu} TE_G$.

Image of this map, i.e., $\mathcal{H}(D):=(\mu\circ D)(p^*TM)\subset TE_G$ is a sub bundle. This plays the role of Horizantal sub bundle when defining connection in the sense of Kobayashi and Nomizu.

This is how we relate the two notions of connection on a principal $G$-bundle $E_G\rightarrow M$, one as a splitting of the Atiyah sequence of vector bundles and the other as a distribution of $TE_G$ satisfying some properties.

Another way I learned from Ralph Cohen's notes (Page number $35$). He does not say in this form, I am interpreting. One can pullback the exact sequence $$0\rightarrow \text{ad}(E_G)\rightarrow \text{At}(E_G)\rightarrow TM\rightarrow 0 $$ of vector bundles over $M$ along the map $p:E_G\rightarrow M$ to get the exact sequence $$0\rightarrow p^*(\text{ad}(E_G))\rightarrow p^*(\text{At}(E_G))\rightarrow p^*(TM)\rightarrow 0 $$ of vector bundles over $P$. Ralph Cohen defines a connection on $p:E_G\rightarrow M$ to be a $G$-equivariant splitting of the above exact sequence of vector bundles. As mentioned above, there is an isomrophism of vector bundles $p^*(\text{At}(E_G))\cong TE_G$, that gives following sequence $$0\rightarrow p^*(\text{ad}(E_G))\rightarrow TE_G\rightarrow p^*(TM)\rightarrow 0 $$ of vector bundles over $P$. A section of above sequence $s:p^*(TM)\rightarrow TE_G$ gives a sub bundle $\mathcal{H}:=s(p^*(TM))\subset TE_G$, giving a distribution. This is compatible with the structure of prinipal bundle, giving a connection in the sense of distribution $\mathcal{H}\subseteq TE_G$.


This is from another reference. So, adding as a different answer.

Appendix A "On principal bundles and Atiyah sequences" in the book Lie groupoids and Lie algebroids in differential geometry by Kirill Mackenzie discuss Atiyah sequence approach of connection on Principal bundles.

AMS review of the book by Antonio Kumpera can be found here

  • $\begingroup$ This is already very standard book and does not need my appreciation but I would like to say this is very well written (I only saw appendix)... $\endgroup$ – Praphulla Koushik Oct 3 '19 at 3:00
  • $\begingroup$ I will be more than happy to discuss about the appendix (any thing else) from the book.. $\endgroup$ – Praphulla Koushik Oct 3 '19 at 3:07

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