Editing Connection (principal bundle) (section)

== Examples ==

=== Maurer-Cartan connection ===
For the trivial principal <math>G</math>-bundle <math>\pi:E \to X</math> where <math>E = G\times X</math>, there is a canonical connection<ref name=":0">{{Cite web |last=Dupont |first=Johan |date=August 2003 |title=Fibre Bundles and Chern-Weil Theory |url=http://www.johno.dk/mathematics/fiberbundlestryk.pdf |archive-url=https://web.archive.org/web/20220331053124/http://www.johno.dk/mathematics/fiberbundlestryk.pdf |archive-date=31 March 2022}}</ref><sup>pg 49</sup><blockquote><math>\omega_{MC} \in \Omega^1(E,\mathfrak{g})</math></blockquote>called the Maurer-Cartan connection. It is defined at a point <math>(g,x) \in G\times X</math> by<blockquote><math>(\omega_{MC})_{(g,x)} = (L_{g^{-1}}\circ \pi_1)_*</math> for <math>x \in X, g \in G</math></blockquote>which is a composition<blockquote><math>T_{(g,x)}E \xrightarrow{\pi_{1*}} T_gG \xrightarrow{(L_{g^{-1}})_*} T_eG = \mathfrak{g}</math></blockquote>defining the 1-form. Note that<blockquote><math>\omega_0 = (L_{g^{-1}})_*: T_gG \to T_eG = \mathfrak{g}</math></blockquote>is the [[Maurer-Cartan form]] on the Lie group <math>G</math> and <math>\omega_{MC} = \pi_1^*\omega_0</math>.

=== Trivial bundle ===
For a trivial principal <math>G</math>-bundle <math>\pi:E \to X</math>, the identity section <math>i: X \to G\times X</math> given by <math>i(x) = (e,x)</math> defines a 1-1 correspondence<blockquote><math>i^*:\Omega^1(E,\mathfrak{g}) \to \Omega^1(X,\mathfrak{g})</math></blockquote>between connections on <math>E</math> and <math>\mathfrak{g}</math>-valued 1-forms on <math>X</math><ref name=":0" /><sup>pg 53</sup>. For a <math>\mathfrak{g}</math>-valued 1-form <math>A</math> on <math>X</math>, there is a unique 1-form <math>\tilde{A}</math> on <math>E</math> such that

# <math>\tilde{A}(X) = 0</math> for <math>X \in T_xE</math> a vertical vector
# <math>R_g^*\tilde{A} = \text{Ad}(g^{-1}) \circ \tilde{A}</math> for any <math>g \in G</math>

Then given this 1-form, a connection on <math>E</math> can be constructed by taking the sum<blockquote><math>\omega_{MC} + \tilde{A}</math></blockquote>giving an actual connection on <math>E</math>. This unique 1-form can be constructed by first looking at it restricted to <math>(e,x)</math> for <math>x \in X</math>. Then, <math>\tilde{A}_{(e,x)}</math> is determined by <math>A</math> because <math>T_{(x,e)}E = ker(\pi_*)\oplus i_*T_xX</math> and we can get <math>\tilde{A}_{(g,x)}</math>by taking<blockquote><math>\tilde{A}_{(g,x)} = R^*_g\tilde{A}_{(e,x)} = \text{Ad}(g^{-1})\circ \tilde{A}_{(e,x)}</math></blockquote>Similarly, the form<blockquote><math>\tilde{A}_{(x,g)} = \text{Ad}(g^{-1}) \circ A_x \circ \pi_*: T_{(x,g)}E \to \mathfrak{g} </math></blockquote>defines a 1-form giving the properties 1 and 2 listed above.

==== Extending this to non-trivial bundles ====
This statement can be refined<ref name=":0" /><sup>pg 55</sup> even further for non-trivial bundles <math>E \to X</math> by considering an open covering <math>\mathcal{U} = \{U_a\}_{a \in I}</math> of <math>X</math> with [[Trivialization (mathematics)|trivializations]] <math>\{\phi_a\}_{a \in I}</math> and transition functions <math>\{g_{ab}\}_{a,b\in I}</math>. Then, there is a 1-1 correspondence between connections on <math>E</math> and collections of 1-forms<blockquote><math>\{A_a \in \Omega_1(U_a,\mathfrak{g}) \}_{a \in I}</math></blockquote>which satisfy<blockquote><math>A_b = Ad(g_{ab}^{-1})\circ A_a + g_{ab}^*\omega_0</math></blockquote>on the intersections <math>U_{ab}</math> for <math>\omega_0</math> the [[Maurer–Cartan form|Maurer-Cartan form]] on <math>G</math>, <math>\omega_0 = g^{-1}dg</math> in matrix form.

==== Global reformulation of space of connections ====
For a principal <math>G</math> bundle <math>\pi: E \to M</math> the set of connections in <math>E</math> is an affine space<ref name=":0" /><sup>pg 57</sup> for the vector space <math>\Omega^1(M,E_\mathfrak{g})</math> where <math>E_\mathfrak{g}</math> is the associated adjoint vector bundle. This implies for any two connections <math>\omega_0, \omega_1</math> there exists a form <math>A \in \Omega^1(M, E_\mathfrak{g})</math> such that<blockquote><math>\omega_0 = \omega_1 + A</math></blockquote>We denote the set of connections as <math>\mathcal{A}(E)</math>, or just <math>\mathcal{A}</math> if the context is clear.

=== Connection on the complex Hopf-bundle ===
We<ref name=":0" /><sup>pg 94</sup> can construct <math>\mathbb{CP}^n</math> as a principal <math>\mathbb{C}^*</math>-bundle <math>\gamma:H_\mathbb{C} \to \mathbb{CP}^n</math> where <math>H_\mathbb{C} = \mathbb{C}^{n+1}-\{0\}</math> and <math>\gamma</math> is the projection map<blockquote><math>\gamma(z_0,\ldots,z_n) = [z_0,\ldots,z_n]</math></blockquote>Note the Lie algebra of <math>\mathbb{C}^* = GL(1,\mathbb{C})</math> is just the complex plane. The 1-form <math>\omega \in \Omega^1(H_\mathbb{C},\mathbb{C})</math> defined as<blockquote><math>\begin{align}
\omega &= \frac{\overline{z}^tdz}{|z|^2} \\
&= \sum_{i=0}^n\frac{\overline{z}_i}{|z|^2}dz_i
\end{align}</math></blockquote>forms a connection, which can be checked by verifying the definition. For any fixed <math>\lambda \in \mathbb{C}^*</math> we have<blockquote><math>\begin{align}
R_\lambda^*\omega &= \frac{\overline{(z\lambda)}^td(z\lambda)}{|z\lambda|^2} \\
&= \frac{
  \overline{\lambda}\lambda\overline{z}^tdz
}{|\lambda|^2\cdot |z|^2}

\end{align}</math></blockquote>and since <math>|\lambda|^2 = \overline{\lambda}{\lambda}</math>, we have <math>\mathbb{C}^*</math>-invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any <math>z \in H_\mathbb{C}</math> we have a short exact sequence<blockquote><math>0 \to \mathbb{C} \xrightarrow{v_z} T_zH_\mathbb{C} \xrightarrow{\gamma_*} T_{[z]}\mathbb{CP}^n \to 0</math></blockquote>where <math>v_z</math> is defined as<blockquote><math>v_z(\lambda) = z\cdot \lambda</math></blockquote>so it acts as scaling in the fiber (which restricts to the corresponding <math>\mathbb{C}^*</math>-action). Taking <math>\omega_z\circ v_z(\lambda)</math> we get

<math>\begin{align}
\omega_z\circ v_z(\lambda) &= \frac{\overline{z}dz}{|z|^2}(z\lambda) \\
&= \frac{\overline{z}z\lambda}{|z|^2} \\
&= \lambda

\end{align}</math>

where the second equality follows because we are considering <math>z\lambda</math> a vertical tangent vector, and <math>dz(z\lambda) = z\lambda</math>. The notation is somewhat confusing, but if we expand out each term<blockquote><math>\begin{align}
dz &= dz_0 + \cdots + dz_n \\
z &= a_0z_0 + \cdots +a_nz_n \\
dz(z) &= a_0 + \cdots + a_n \\
dz(\lambda z) &= \lambda\cdot (a_0 + \cdots + a_n) \\
\overline{z} &= \overline{a_0} + \cdots + \overline{a_n}

\end{align}</math></blockquote>it becomes more clear (where <math>a_i \in \mathbb{C}</math>).