Modern differential geometry provides a unified picture for different aspects of physics. In this blog article I would like to introduce three kinds of physical subjects relating to connection on a fibre bundle, namely gauge field, general relativity and theory of quantum Hall effect. The ‘gauge field’ part is among the appendices of my undergraduate thesis.
Gauge field
Mathematically, gauge fields on a continuous space is described by connections on a vector bundle. Here we briefly summarize the mathematical notions see how it can be related to a gauge theory.
Geometric perspective
The definition of a vector bundle which we take as our starting point can be found in standard textbooks^{1}. Suppose $E$ is a vector bundle on $M$ and $\Gamma\left(E\right)$ is the set of smooth sections of $E$ on $M$, a connection over a vector bundle $E$ is defined as a map
\[D:\Gamma\left(E\right)\rightarrow \Gamma\left(E\otimes T^*\left(M\right)\right),\]which satisfies

For any $s_1,s_2\in\Gamma\left(E\right)$,
\[D\left(s_1+s_2\right)=Ds_1+Ds_2.\] 
For $s\in\Gamma\left(E\right)$ and any $\alpha\in C^\infty \left(M\right)$,
\[D\left(s\alpha\right)=s\otimes\mathrm{d}\alpha+\left(Ds\right)\alpha.\]
Given a smooth tangent vector field $X$ on $M$, define the covariant derivative of section $s$ along $X$ as
\[D_Xs=\left\langle X,Ds\right\rangle.\]We can extend the notion of connection to more general cases. Define $\Omega_M^p\left(E\right)=\Gamma\left(E\otimes \bigwedge^p T^*M\right)$. Connection can then defined as a map
\[D:\Omega_M^p\left(E\right)\rightarrow\Omega_M^{p+1}\left(E\right),\]such that

For any $s_1,s_2\in\Omega_M^p\left(E\right)$,
\[D\left(s_1+s_2\right)=Ds_1+Ds_2.\] 
For $s\in\Omega_M^0\left(E\right)$ and a $p$form $\alpha\in \Omega_M^p$,
\[D\left(s\otimes\alpha\right)=s\otimes\mathrm{d}\alpha+D\left(s\right)\wedge\alpha.\]
For $p=0$, this definition reduces to the former one.
We can further consider $\Omega_M^p\left(\operatorname{End}\left(E\right)\right)$ which maps a section in $\Omega_M^r\left(E\right)$ naturally to a section in $\Omega_M^{p+r}\left(E\right)$. By ‘naturally’ we mean that for $s\in\Omega_M^0\left(E\right)$ and a $p$form $\alpha\in \Omega_M^p$ we have
\[\phi\left(s\otimes\alpha\right)=\phi\left(s\right)\wedge \alpha.\]It is plausible since the endomorphism is linear. For $\phi\in\Omega_M^p\left(\operatorname{End}\left(E\right)\right)$ and $s\in\Omega_M^0\left(E\right)$, define $D\phi\in\Omega_M^{p+1}\left(\operatorname{End}\left(E\right)\right)$
\[\left(D\phi\right)s=D\left(\phi s\right)\left(1\right)^p\phi\left(Ds\right).\]Now comes the curvature. The curvature $F\in\Omega_M^2\left(\operatorname{End}\left(E\right)\right)$ is defined such that for $s\in\Omega_M^0\left(E\right)$,
\[D^2s = Fs.\]Bianchi identity follows directly that
\[\left(DF\right)s=D\left(Fs\right)F\left(Ds\right)=D^3sD^3s=0,\]or
\[DF=0.\]Here we have used that acting on $\Omega_M^p\left(E\right)$, $F$ is also equivalent to $D^2$.
Frame perspective
Note that all the definitions above are geometric objects, that is, they are independent of ‘coordinates’ or ‘physical’. To make it convenient for calculation, introduce a local frame field of the vector bundle $\hat{s}_i\in\Omega_M^0\left(E\right)$ such that they are linearly independent and expand the vector bundle locally. To write explicitely the coefficients of the connection, we can define $\Gamma{^i}{_{j\mu}}$ as
\[D\hat{s}_i=\hat{s}_j\otimes\Gamma{^j}{_{i\mu}}\mathrm{d}x^\mu.\]Now we can introduce a matrix valued oneform $\omega\in\Omega_M^1\left(\operatorname{End}\left(E\right)\right)$ called connection form as
\[\omega = \Gamma{^j}{_{i\mu}}\mathrm{d}x^\mu.\]For notational convenience, we use $s$ to refer to either a section $s=s^i\hat{s}_i\otimes \alpha\in\Omega_M^p\left(E\right)$ where $\alpha$ is a $p$form or the vector formed by its components $s^i$. The connection is then
\[D\left(s^i\hat{s}_i\otimes \alpha\right)=\left(\partial_\mu s^i+\Gamma{^i}{_{j\mu}}s^j\right)\hat{s}_i\otimes\mathrm{d}x^\mu\wedge \alpha,\]or in a notational simple way
\[\begin{equation} Ds=\mathrm{d}s+\omega\wedge s. \label{app1_connection coordinate form} \end{equation}\]While for $\phi\in\Omega_M^p\left(\operatorname{End}\left(E\right)\right)$ whose components can be viewed as matrices given a local frame, we have
\[\phi s = \phi \wedge s,\]where the components are matrix producting a vector. Setting $s\in\Omega_M^0\left(E\right)$, acting $D$ on $\phi$ gives
\[\left(D\phi\right) s=D\left(\phi s\right)\left(1\right)^p\phi\left(Ds\right)=\mathrm{d}\left(\phi s\right)+\omega\wedge\phi s\left(1\right)^p\left(\phi\wedge\mathrm{d}s+\phi\wedge\omega s\right)\]and hence
\[\begin{equation} D\phi = \mathrm{d}\phi +\omega\wedge \phi  \left(1\right)^p\phi\wedge\omega. \label{app1_covariant derivative on phi} \end{equation}\]Curvature is now
\[\begin{equation} F = \mathrm{d}\omega+\omega\wedge\omega, \label{app1_curvature coordinate form} \end{equation}\]and the Bianchi identity has the form
\[\begin{equation} \mathrm{d}F+\omega\wedge FF\wedge\omega = 0. \label{app1_Bianchi identity in frame} \end{equation}\]If we change from one local frame to another, yielding the coordinate transformation $s’=Vs$, coordinate equation (\ref{app1_connection coordinate form}) should change as
\[Ds'=VDs,\]that is,
\[\mathrm{d}s'+\omega'\wedge s' = V\mathrm{d}\left(V^{1}s'\right)+V\omega\wedge V^{1}s'.\]Comparing the coefficients we have
\[\begin{equation} \omega'=V\left(\partial_\mu\mathrm{d}x^\mu+\omega\right)V^{1}. \label{app1_frame transformation of connection form} \end{equation}\]By definition, the curvature $F$ is `physical’ therefore we expect a natural transformation
\[F'=VFV^{1}.\]We can check this by
\[\begin{aligned} F'&=\mathrm{d}\omega'+\omega'\wedge\omega'\\ &=\partial_\nu V\mathrm{d}x^\nu\wedge\left(\mathrm{d}x^\mu\partial_\mu+\omega\right)V^{1}V\left(\mathrm{d}x^\mu\partial_\mu+\omega\right)\wedge\mathrm{d}x^\nu\partial_\nu V^{1}\\ &\qquad{} +V\partial_\mu V^{1} V\partial_\nu V^{1}\mathrm{d}x^\mu\wedge\mathrm{d}x^\nu + V\partial_\mu V^{1}V\mathrm{d}x^\mu\wedge\omega V^{1}+V\omega\wedge \mathrm{d}x^\mu \partial_\mu V^{1}+VFV^{1}\\ &=VFV^{1}. \end{aligned}\]Gauge field
With the basic knowledge of connections on a vector field, we can make the substitution
\[\omega\rightarrow \mathrm{i}gA_\mu\mathrm{d}x^\mu,\\ F\rightarrow \frac{\mathrm{i}g}{2}F_{\mu\nu}\mathrm{d}x^\mu\wedge\mathrm{d}x^\nu.\]Therefore, equation (\ref{app1_connection coordinate form}) becomes
\[D_\mu = \partial_\mu \mathrm{i}g A_\mu.\]Equation (\ref{app1_curvature coordinate form}) becomes
\[F_{\mu\nu}=\partial_\mu A_\nu\partial_\nu A_\mu\mathrm{i}g\left[A_\mu,A_\nu\right].\]Equation (\ref{app1_frame transformation of connection form}) becomes
\[A_\mu'=V\left(A_\mu+\frac{\mathrm{i}}{g}\partial_\mu\right)V^{1}.\]Covariant derivative acting on $F_{\mu\nu}$ should be equation (\ref{app1_covariant derivative on phi})
\[D_\rho F_{\mu\nu}=\partial_\rho F_{\mu\nu}\mathrm{i}g\left[A_\rho, F_{\mu\nu}\right].\]Finally we have Bianchi identity (\ref{app1_Bianchi identity in frame})
\[D_{\left[\rho\right.}F_{\left.\mu\nu\right]}=0,\]which yields the sourceless Maxwell equations.
General relativity
Mathematics for general relativity is of tangent and cotangent bundles, which is a special case for that of gauge field.
To be continued.
Berry connection and Berry phase
Berry connection is a connection on principal $\mathrm{U}(1)$bundle.
Principal bundles and connection
Berry phase and quantum Hall effect
To be continued.

S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry. WORLD SCIENTIFIC, 1999. ↩