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

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

Proof. Exercise 6, hom11zadaci

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 $x$.