# Principal $G\dash$Bundles and Connections (Monday, January 25)

:::{.definition title="Principal Bundles"}
Let $G$ be a (possibly disconnected) Lie group.
Then a **principal $G\dash$bundle** $\pi:P\to X$ is a space admitting local trivializations $h_u: \pi ^{-1} (U) \to G \cross U$ such that the transition functions are given by left multiplication by a continuous function $t_{UV}: U \intersect V \to G$.

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:::{.remark}
Setup: 
we'll consider $TX$ for $X\in \Mfd_\Sm$, and let $g$ be a metric on the tangent bundle given by 
\[
g_p: T_pX^{\tensor 2} \to \RR
,\] 
a symmetric bilinear form with $g_p(u, v) \geq 0$ with equality if and only if $v=0$.

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:::{.definition title="The Frame Bundle"}
Define $\Frame_p(X) \da \ts{\text{bases of } T_p X}$, and $\Frame(X) \da \Union_{p\in X} \Frame_p(X)$.
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:::{.remark}
More generally, $\Frame (\bundle{E})$ can be defined for any vector bundle \( \mathcal{E} \), so $\Frame(X) \da \Frame(TX)$.
Note that $\Frame(X)$ is a principal $\GL_n(\RR)\dash$bundle where $n\da \rank(\mathcal{E})$.
This follows from the fact that the transition functions are fiberwise in $\GL_n(\RR)$, so the transition functions are given by left-multiplication by matrices.
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:::{.remark title="Important"}
A principal $G\dash$bundle admits a $G\dash$action where $G$ acts by *right* multiplication:
\[
P \cross G \to P \\
( (g, x), h) \mapsto (gh, x)
.\]
This is necessary for compatibility on overlaps.
**Key point**: the actions of left and right multiplication commute.
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:::{.definition title="Orthogonal Frame Bundle"}
The **orthogonal frame bundle** of a vector bundle \( \mathcal{E}  \) equipped with a metric $g$ is defined as $\OFrame_p(\bundle{E}) \da \ts{\text{orthonormal bases of } \mathcal{E}_p}$, also written $O_r(\RR)$ where $r \da \rank( \mathcal{E})$. 
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:::{.remark}
The fibers $P_x \to \ts {x}$ of a principal $G\dash$bundle are naturally **torsors** over $G$, i.e. a set with a free transitive $G\dash$action.
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:::{.definition title="Hermitian metric"}
Let \( \mathcal{E}\to X  \) be a complex vector bundle.
Then a **Hermitian metric** is a hermitian form on every fiber, i.e. 
\[
h_p: \mathcal{E}_p \cross \overline{\mathcal{E}_p } \to \CC
.\]
where $h_p(v, \bar{v} ) \geq 0$ with equality if and only if $v=0$.
Here we define \( \overline{\mathcal{E}_p}  \) as the fiber of the complex vector bundle \( \overline{\mathcal{E}}  \) whose transition functions are given by the complex conjugates of those from \( \mathcal{E}  \).
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:::{.remark}
Note that \( \mathcal{E}, \overline{\mathcal{E}}   \) are genuinely different as complex bundles.
There is a *conjugate-linear* map given by conjugation, i.e. $L(cv) = \bar{c} L(v)$, where the canonical example is 
\[
\CC^n &\to \CC^n \\
(z_1, \cdots, z_n) &\mapsto (\bar {z_1}, \cdots, \bar {z_n})
.\]

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:::{.definition title="Unitary Frame Bundle"}
We define the **unitary frame bundle** $\UFrame(\bundle{E}) \da \Union_p \UFrame(\bundle{E})_p$, where at each point this is given by the set of orthogonal frames of \( \mathcal{E}_p  \) given by $(e_1, \cdots, e_n)$ where $h(e_i , \bar {e_j} ) = \delta_{ij}$.
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:::{.remark}
This is a principal $G\dash$bundle for $G = U_r(\CC)$, the invertible matrices $A_{/\CC}$ satisfy $A \overline{A}^t = \id$.
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:::{.example title="of more principal bundles"}
For $G=\ZZ/2\ZZ$ and $X= S^1$, the Möbius band is a principal $G\dash$bundle:

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:::{.example title="more principal bundles"}
For $G=\ZZ/2\ZZ$, for any (possibly non-oriented) manifold $X$ there is an **orientation principal bundle** $P$ which is locally a set of orientations on $U$, i.e. 
\[
P\da \ts{(x, O) \st x\in X,\, O \text{ is an orientation of }T_p X}
.\]
Note that $P$ is an oriented manifold, $P\to X$ is a local isomorphism, and has a canonical orientation. (?)
This can also be written as $P = \Frame(X) / \GL_n^+(\RR)$, since an orientation can be specified by a choice of $n$ linearly independent vectors where we identify any two sets that differ by a matrix of positive determinant.
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:::{.definition title="Associated Bundles"}
Let $P\to X$ be a principal $G\dash$bundle and let $G\to \GL(V)$ be a continuous representation.
The **associated bundle** is defined as 
\[
P\cross_G V = \ts{(p, v)\st p\in P,\, v\in V} / \sim && \text{where } (p, v) \sim (pg, g ^{-1} v)
,\]
which is well-defined since there is a right action on the first component and a left action on the second.
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:::{.example title="?"}
Note that $\Frame(\bundle{E})$ is a $\GL_r(\RR)\dash$bundle and the map $\GL_r(\RR) \mapsvia{\id} \GL(\RR^r)$ is a representation.
At every fiber, we have $G \cross_G V = (p, v)/\sim$ where there is a unique representative of this equivalence class given by $(e, pv)$.
So $P\cross_G V_p \to \ts{p} \cong V_x$.


:::{.exercise title="?"}
Show that $\Frame(\bundle{E}) \cross_{\GL_r(\RR)} \RR^r \cong \mathcal{E}$.
This follows from the fact that the transition functions of $P \cross_G V$ are given by left multiplication of $t_{UV}: U \intersect V \to G$, and so by the equivalence relation, $\im t_{UV} \in \GL(V)$.

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:::{.remark}
Suppose that $M^3$ is an oriented Riemannian 3-manifold.
Them $TM\to \Frame (M)$ which is a principal $\SO(3)\dash$bundle.
The universal cover is the double cover $\SU(2) \to \SO(3)$, so can the transition functions be lifted?
This shows up for spin structures, and we can get a $\CC^2$ bundle out of this.
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