# Zoran Skoda hom11lec9

Srijeda 25. 1. u 16 sati. Na početku sam ukratko opisao sadržaj lec7 i lec8. Slijedeća lec10. Coursepage hom11connections. Other lectures 1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,zadaci. Ovdje uglavnom slijedimo Postnikovljeve lekcije iz geometrije (semestar 4 i sem. 6).

## Forms of connections (continuation)

Given a fiber bundle $E\to M$, an Ehresmann connection is given by a smooth field of horizontal subspaces $T^H_e E\subset T_e E, e\in E$ in the tangent bundle $T P$ (horizontal means $T^H_e E\oplus T^V_e E = T_e E$ for all $e\in E$. In a local trivialization over an open chart with coordinates $x^i$ and with local coordinate $a^i$ in a neighborhood in a typical fiber, one can choose a set of forms $\theta^i$, which are linearly independent at every point of the form

$\theta^i = f^i_j d a^i + g^i_k d x^k$

such that $T^H E = Ann(\theta^1,\ldots,\theta^n)$. In the case of principal $G$-bundles $i = 1,\ldots, n^2$ and $k = 1,\ldots, m$; and in the case of vector bundles $i = 1,\ldots, n$ and $k = 1,\ldots, m$. The annihilator does not change by an invertible linear transformation given by multiplying with the inverse of the matrix $(f^i_j)$, hence we can always choose (after renaming)

$\theta^i = d a^i + g^i_k d x^k$

For a smooth field of horizontal subspaces, to be an Ehresman connection, in the case of principal bundles there is an equivariance condition imposed on $T^H E$, while in the case of vector bundles one needs the linearity of $g^i_k$ in the fiber, i.e.

$g^i_k = \Gamma^i_{k j} a^j$

Note that the order between $k$ and $j$ is not fully uniform in the literature. We also write

$\omega^i_j := \Gamma^i_{k j} d x^k$

i.e.

$\theta^i = d a^i + \omega^i_j a^j$

and we say that $\omega^i_j$ is the 1-form of the connection (and one often omits 1 in the name). Sometimes, more rarely, one also says that $\theta^i$ are the 1-forms of connection.

#### Gauge transformations

Consider the cover of the manifold by bundle charts and consider the intersection of two elements of the cover. On the intersection, we consider the coordinate changes

$a^i = \phi^i_{i'}(\mathbf{x'}) a^{i'},\,\,\,\,\,\,x^k = x^k(\mathbf{x}')$

As usual we apply the chain rule to find out the changes of differentials involved in the expressions for $\theta^i = d a^i + \Gamma^{i}_{k j} a^j d x^k$ in both charts; e.g.

$d x^k = \frac{\partial x^k}{\partial x^{k'}} d x^{k'}, \,\,\,\,k,k' =1,\ldots,n$
$d \phi^i_{j'} = \frac{\partial \phi^i_{j'}}{\partial x^{k'}} d x^{k'}$

After a small calculation, which was sketched in the lecture, we arrive at the transformation law

$\Gamma^{i'}_{k' j'} = \phi^{i'}_i \phi^j_{j'} \frac{\partial x^k}{\partial x^{k'}} \Gamma^i_{k j} + \phi^{i'}_i \frac{\partial \phi^i_{j'}}{\partial x^{k'}}$
$\omega' = \phi^{-1} \omega \phi + \phi^{-1} d \phi$

or if we label the neighborhoods of the cover by $\alpha,\beta...$

$\stackrel{(\beta)}\omega = \phi^{-1}_{\beta\alpha}\stackrel{(\alpha)}\omega\phi_{\beta\alpha} + \phi^{-1}_{\beta\alpha} d\phi_{\beta\alpha}$

These changes of 1-forms of connection are called gauge transformations.

## Covariant derivative $\nabla_X$

#### Construction from the Ehresmann connection

Let $X$ be a vector field in the base of the vector bundle $V\to M$ with a connection. A covariant derivative of a smooth section $s:M\to V$ along the vector field $X\in \Gamma T M$ is a smooth section $\nabla_X s \in \Gamma T M$ given by the formula

$(\nabla_X s)(b) = lim_{t\to 0} \frac{s(u(t)) - v(t)}{t},$

where $u: I\to M$ is the integral curve of the vector field $X$ defined in the neighborhood of $b$ (i.e. $\frac{d u}{d t} = X_{u(t)}$ for $t\in I$ for some interval $I$ around $0$) with the initial condition $b = u(0)$, and $v : I\to V$ is the horizontal lift of $u$ i.e. $p(v(t)) = u(t)$, $t\in I$ and $\frac{d v}{d t} \in T^H V$ for all $t$; with the initial condition $s(b) = v(0)$.

#### General properties and axioms

A covariant derivative rule is an assignment $X\mapsto \nabla_X: \Gamma \xi\to \Gamma \xi$, where $\xi = (p:V\to M)$, satisfying $C^\infty(M)$-linearity in $X$

$\nabla_{f X + g Y} = f\nabla_X + g\nabla_Y$

for all $X,Y\in\Gamma T M$, $f,g\in C^\infty(M)$, $\mathbf{R}$-linearity in the argument $s$, and such that

$\nabla_X (f s) = (X f) s + f \nabla_X(s), \,\,\,\,\,\,f\in C^\infty(M), s\in \Gamma\xi.$

## Covariant differentiation

Given a smooth bundle $\xi = (E\to M)$, a covariant differentiation is any $\mathbf{R}$-linear map $\nabla:\Gamma \xi\to \Gamma (T^* M\otimes \xi)$ such that $\nabla(f s) = df\otimes s + f\nabla s$ where $f\in C^\infty(M)$ and $s\in \Gamma \xi$. (We sometimes write sloppy $\Gamma E$ meaning $\Gamma\xi$).

It appears that such differentiations are in 1-1 correspondence with covariant derivatives coming from connections. Namely, let $e_1,\ldots,e_n$ be a basis of $\Gamma_U \xi$ above some contractible neighborhood $U\subset M$. Then $\nabla e_j = \sum_{i,k} \Gamma^i_{k j} d x^k\otimes e_i$ for some functions $\Gamma^i_{k j} = \Gamma^i_{k j}(x)$ on $M$. It appears that these functions, under the coordinate changes, change as the Christoffel symbols for some connection. One may also write $\nabla e_j = \sum_i \omega^i_j \otimes e_j$. Then for the arbitrary section $s = \sum_i s^i e_i \in \Gamma_U \xi$ we have (by $\mathbf{R}$-linearity and Leibniz rule) $\nabla s = \sum_i (d s^i + \omega^i_j s^j)\otimes e_i$.

## Parallel transport, holonomy and monodromy

Let $\pi:E\to M$ be a principal $G$-bundle with an Ehresmann connection and $u: I = [0,t_0]\to M$ a regular smooth curve with $u(0) = x$ and $u(t_0) = y$. Then for each $e\in \pi^{-1}(e)$, there is a unique horizontal lift of $u$, i.e. a smooth curve $v:I\to E$ such that $\pi\circ v = u$ and at each point the tangent vector to $v$ is horizontal, i.e. belongs to the horizontal subspace $T^H_{v(0)} E\subset T_{v(0)} E$. Define $\Pi_u(e)=v(t_0)\in \pi^{-1}(y)$. The rule $\Pi_u :E_x\to E_y : e\mapsto \Pi_u(e)$ is the parallel transport along curve $u$.

By the equivariance of the field of horizontal subspaces, it follows that if $x=y$, i.e. $u$ is a smooth loop, $\Pi_u$ is the right shift by a unique element $g\in G$, hence $\Pi_u$ can be considered an element in $G$. The holonomy subgroup at point $x$ is the group of all $\Pi_u\in G$ for all smooth loops $u$ at $x$. It is a Lie subgroup of $G$. Knowing the holonomy subgroup of a connection often tells information about the bundle. Each bundle, as connections vary, can have only some holonomy subgroups, not all subgroups of $G$.

In the case that the connection is flat (curvature is zero, see below), the parallel transport does not depend on a choice of a curve up to homotopy. In other words, the parallel transport gives a representation of a fundamental group. Such representations are called monodromies. Each such representation has a unique covering space associated. We say that we have a local system.

## Curvature and curvature form

Theorem. The covariant differentiation $\nabla:\Gamma E\to \Gamma (T^*M\otimes E) \cong \Gamma T^* M\otimes_{C^\infty(M)} \Gamma E \cong \Omega^1(M)\otimes_{C^\infty(M)} \Gamma E$ canonically extends to higher forms, i.e. there is a unique $\mathbf{R}$-linear map $\tilde\nabla : \Gamma E\otimes \Omega^\bullet M \to \Gamma E\otimes \Omega^{\bullet+1} M$, such that $\tilde\nabla|_{\Gamma E} = \nabla$ and such that for all $s\in \Gamma E$ and homogeneous $\omega\in\Omega^\bullet(M)$, the Leibniz rule holds

$\tilde\nabla(\omega\otimes s) = d\omega\otimes s + (-1)^{deg \omega} \omega\otimes \nabla(s).$

The curvature 2-form is the map $\Omega = \tilde{\nabla}\circ\nabla : \Gamma E\to \Gamma(\wedge^2 T^* M\otimes E) = \Omega^2(M)\otimes\Gamma E$.

Proposition. $\Omega = \tilde{\nabla}\circ\nabla$ is a $C^\infty(M)$-linear map.

Proof. Give $f\in C^\infty(M)$ and $S\in \Gamma E$, we compute

$\array{ (\tilde{\nabla}\circ \nabla)(f s) &=& \tilde\nabla(d f \otimes s + f\nabla(s)) \\ &=& d d f\otimes s - d f\otimes \nabla(s) + d f \otimes \nabla(s) + f\tilde\nabla(\nabla(s)) \\ & =& f (\tilde{\nabla}\circ \nabla)(s) }$

Zadatak 5. Neka su $F_1\to M$, $F_2\to M$ dva glatka vektorska svežnja i $\Gamma F_1$ i $\Gamma F_2$ prostori glatkih prereza tih svežnjeva nad parakompaktnom glatkom mnogostrukošću $M$, koje možemo promatrati kao $C^\infty(M)$-module. Neka je $D:\Gamma F_1\to \Gamma F_2$ proizvoljno preslikavanje koje ima jako svojstvo da je $C^\infty(M)$-linearno (ne samo $\mathbf{R}$-linearno). Dokaži da postoji dobro definirano preslikavanje $D_0 : F_1\to F_2$ vektorskih svežnjeva nad $M$ takvo da za svaki prerez $s\in\Gamma F_1$ i za svaku točku $x\in M$ vrijedi $D(s)(x) = D_0(s(x))$.

According to the exercise 5 and the fact that $\tilde\nabla\circ\nabla$ is $C^\infty(M)$-linear, we infer that there exist a bundle map $R:E\to \wedge^2 T^*M\otimes E$, such that on all sections $\tilde\nabla\circ\nabla = R\circ$. We say that $R$ is the tensor of curvature. Then, overf a contractible neighborhood $U$ with $C^\infty(M)$ basis $\{e_i\}_i$ of the module of sections of $\Gamma E$, define $\Omega^i_j$ by

$R\circ e_j = \sum_i \Omega^i_j \otimes e_i$

Then

$\Omega^i_j = d \omega^i_j + \omega^i_p\wedge \omega^p_j$

or, in matrix form, $\Omega = d\omega +\omega\wedge\omega$. Here, as before, $\omega^i_j = \sum_k \Gamma^i_{k j} d x^k$. The Maurer-Cartan equation is the statement that $\Omega = 0$.

### Ambrose-Singer’s theorem

Given a Lie group $G$ and a principal $G$-bundle $P\to M$ with a connection, and a point $p\in P$, the Lie algebra of the group of holonomy at $x$ is spanned as a vector space by all possible expressions of the form $\Omega_x(X,Y)$ where $X,Y$ are tangent vectors at $x$ and $\Omega_x$ is the curvature 1-form at $x$.

Last revised on April 14, 2012 at 14:26:28. See the history of this page for a list of all contributions to it.