# Characterisation of (integrable) connections on (trivial) principal bundle

Let $$M$$ be a manifold. Let $$G$$ be a Lie group and $$\mathfrak{g}$$ be its Lie algebra.

Let $$P(M,G)$$ be a principal bundle. Recall that, a connection on $$P(M,G)$$ is a distribution $$\mathcal{H}\subseteq TP$$ satisfying certain conditions. Equivalently, a $$\mathfrak{g}$$-valued differential forms on $$P$$ satisfying certain conditions.

Question : Is there a characterization of connections on $$P(M,G)$$; in the sense, a one-one correspondence between the set of connections on $$P(M,G)$$ and some "well-described set"?

Question : Can we say something if we restrict to the case of integrable connections?

Questions : Can we say something when the bundle is trivial bundle? A characterization for integrable connections on trivial principal bundle?

Edit : May be it is already clear but, I want to specify that I am seeing two connections $$\omega, \omega’$$ to be same if there exists an isomorphism $$\varphi :P\rightarrow P’$$ Of vector bundles such that $$\varphi^*\omega’=\omega$$, that is the pullback of $$\omega’$$ is equal to $$\omega$$.

• Connections are affine over $\Omega^1(M;\mathfrak g)$, and in particular form an affine space. Integrable connections are the same thing as flat conns, which are the zeroes of the curvature map $F_A$. Such a map gives rise to a holonomy map $\pi_1 M \to G$. The space of all flat connections on $P$ carries the action of the automorphism group of $P$, and the space of homomorphisms to $G$, the action of conjugation. After quotienting this descends to a bijection from the space of flat conns to the subset of $\text{Hom}(\pi_1 M, G)$ so that $\tilde M \times_{\pi_1 M} G \cong P$ as G-bundles. – mme May 3 '20 at 17:53
• If your question is how to tell what the topological type of the bundle $\tilde M \times_{\pi_1 M} G$ is given the homomorphism $\rho: \pi_1 M \to G$, you will probably first want to know how to determine the isomorphism classes of principal G-bundles over M, and then to determine it by hand for your representation $\rho$... in particular it's not hard if $G = U(1)$! – mme May 3 '20 at 17:56
• Atleast for trivial Principal $G$ bundles a partial answer to your question is mentioned in the Theorem 1 in arxiv.org/pdf/1003.4485.pdf. Also for non trivial Principal $G$ bundles a partial answer can be found in the section "Idea" in ncatlab.org/nlab/show/connection+on+a+bundle where you may find a one-one correspondence(not sure about 1 direction) between connections on Principal $G$ bundles over $M$ and an appropriate subset of functors from Path groupoid of $M$ to the Atiyah Lie Groupoid of the principal $G$ bundle. – Adittya Chaudhuri May 6 '20 at 21:17
• @AdittyaChaudhuri +1 for Theorem $1$ that gives a one-one correspondence between $\mathfrak{g}$-valued $1$-forms on $M$ and the set of connections on the trivial principal bundle $M\times G\rightarrow M$... I do not understand the other half of the theorem that says there is a one one correspondence between the above mentioned set and the set of "smooth" functors $\mathcal{P}_1(M)\rightarrow G$.. I never understood what is smooth structure on $\mathcal{P}_1(M)$.. Do you want to shre your thoughts on smoothness of these functors? – Praphulla Koushik May 7 '20 at 3:51
• :) See if you can make it as an answer by adding as many details as you can.. – Praphulla Koushik May 7 '20 at 5:04

This is not an answer. This is in response to Mike Miller's comment.

Let $$M$$ be a manifold, $$\tilde{M}$$ to be its associated universal cover (a simply connected covering space over $$M$$). I do not understand why they are not assuming $$M$$ to be connected. I am assuming $$M$$ is a connected manifold.

The following result is from the book Differential Geometry - Bundles, Connections, Metrics and Curvature by Clifford Henry Taubes.

Theorem $$13.2$$ (classification theorem for flat connections) says that, there is a bijection between the sets $$\mathcal{F}_{M,G}$$ and the set $$\text{Hom}(\pi_1(M),G)/G$$ where,

1. $$\mathcal{F}_{M,G}$$ denote the set of equivalence classes of pairs $$(P,A)$$, where $$P\rightarrow M$$ is a principal $$G$$ bundle, and $$A$$ is a flat connection on $$P(M,G)$$. Two pairs $$(P,A)$$ and $$(P',A')$$ are equivalent, if there is an isomorphism of principal $$G$$-bundles $$(\varphi,1_M):(P,M)\rightarrow (P',M)$$ such that $$\varphi^*A'=A$$ (pullback of connection $$A'$$ is the connection $$A$$).
2. $$\text{Hom}(\pi_1(M),G)/G$$ denote the set of equivalence classes of group homomorphisms $$\pi_1(M)\rightarrow G$$. Two morphisms $$f_1:\pi_1(M)\rightarrow G$$ and $$f_2:\pi_1(M)\rightarrow G$$ are equivalent if there exists $$g\in G$$ such that $$f_1=gf_2g^{-1}:\pi_1(M)\rightarrow G$$.

The bijection $$\mathcal{F}_{M,G}\rightarrow \text{Hom}(\pi_1(M),G)/G$$ is given as follows:

• given a principal bundle $$P(M,G)$$ with flat connection $$A$$, we get a group homomorphism $$\pi(M)\rightarrow G$$. Its equivalence class gives an element in $$\text{Hom}(\pi_1(M),G)/G$$.
• Let $$\rho:\pi_1(M)\rightarrow G$$ be a representative of an element in $$\text{Hom}(\pi_1(M),G)/G$$. Consider the trivial principal $$G$$-bundle $$\tilde{M}\times G\rightarrow \tilde{M}$$. The map $$\rho:\pi_1(M)\rightarrow G$$ given an action of $$\pi_1(M)$$ on $$G$$, which in turn gives an action of $$\pi_1(M)$$ on $$\tilde{M}\times G$$. Thus, trivial principal bundle $$\tilde{M}\times G\rightarrow \tilde{M}$$ induce $$(\tilde{M}\times G)/\pi_1(M)\rightarrow M$$. Thus, we get a principal $$G$$-bundle over $$M$$, which we denote by $$\tilde{M}\times_{\rho}G\rightarrow M$$. It turns out that, there exists a $$\mathfrak{g}$$-valued $$1$$-form on $$\tilde{M}\times_{\rho}G\rightarrow M$$ whose pullback to $$\tilde{M}\times G\rightarrow \tilde{M}$$ is the canonical connection on the trivial bundle. It turns out that this $$\mathfrak{g}$$-valued $$1$$-form on $$\tilde{M}\times_{\rho}G\rightarrow M$$ is a flat connection on the principal bundle $$\tilde{M}\times_{\rho}G\rightarrow M$$. Thus, we get a principal bundle $$(P_\rho,A_{\rho})$$. Take its equivalence class to get an element in $$\mathcal{F}_{M,G}$$.

It is not clear how does this answer the question:

Given a principal bundle $$P\rightarrow M$$, how does one know for what $$\rho:\pi_1(M)\rightarrow G$$, do we get that that there is an isomorphism of principal bundle $$P\cong \tilde{M}\times_{\rho}G$$?

I am also interested in only "different" connections in the sense if two connections on $$(P,M)$$ are related by an isomorphism, in the sense of pullbacks, then I am calling these to be same.

• If anyone wants to see more details, who do not have the book, I can give more details... – Praphulla Koushik May 8 '20 at 19:06
• It sounds like the missing piece for you is this image. As I said in my comment: $\rho$ is in the image of $\mathcal F_P$ iff the $G$-bundle $(\widetilde M \times_\rho G))$ is isomorphic to $P$ as a G-bundle, where the subscript $\rho$ indicates we identify $(x, g) \sim (\gamma x, \rho(\gamma) g)$ for all $\gamma, x, g$, thinking of $\gamma \in \pi_1 M$ as a deck transformation. This is as explicit as you will ever get in full generality. If you have a specific bundle over a specific manifold (or your group G is not complicated) you may be able to get better answers by ad hoc techniques. – mme May 8 '20 at 19:18
• @MikeMiller I have clarified that part.. Do you have a procedure to write this down explicitly if I have a principal bundle over a manifold when the group G is reasonably good.. Any reference is welcome.. I enjoyed reading chapter 13 of Taubes book.. – Praphulla Koushik May 9 '20 at 5:43