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

θ i=f j ida i+g k idx k \theta^i = f^i_j d a^i + g^i_k d x^k

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

θ i=da i+g k idx k \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 HET^H E, while in the case of vector bundles one needs the linearity of g k ig^i_k in the fiber, i.e.

g k i=Γ kj ia j g^i_k = \Gamma^i_{k j} a^j

Note that the order between kk and jj is not fully uniform in the literature. We also write

ω j i:=Γ kj idx k \omega^i_j := \Gamma^i_{k j} d x^k

i.e.

θ i=da i+ω j ia j \theta^i = d a^i + \omega^i_j a^j

and we say that ω j i\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 θ i\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=ϕ i i(x)a i,x k=x k(x) 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 θ i=da i+Γ kj ia jdx k\theta^i = d a^i + \Gamma^{i}_{k j} a^j d x^k in both charts; e.g.

dx k=x kx kdx k,k,k=1,,n d x^k = \frac{\partial x^k}{\partial x^{k'}} d x^{k'}, \,\,\,\,k,k' =1,\ldots,n
dϕ j i=ϕ j ix kdx k 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

Γ kj i=ϕ i iϕ j jx kx kΓ kj i+ϕ i iϕ j ix k \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'}}
ω=ϕ 1ωϕ+ϕ 1dϕ \omega' = \phi^{-1} \omega \phi + \phi^{-1} d \phi

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

ω(β)=ϕ βα 1ω(α)ϕ βα+ϕ βα 1dϕ βα \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 X\nabla_X

Construction from the Ehresmann connection

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

( Xs)(b)=lim t0s(u(t))v(t)t, (\nabla_X s)(b) = lim_{t\to 0} \frac{s(u(t)) - v(t)}{t},

where u:IMu: I\to M is the integral curve of the vector field XX defined in the neighborhood of bb (i.e. dudt=X u(t)\frac{d u}{d t} = X_{u(t)} for tIt\in I for some interval II around 00) with the initial condition b=u(0)b = u(0), and v:IVv : I\to V is the horizontal lift of uu i.e. p(v(t))=u(t)p(v(t)) = u(t), tIt\in I and dvdtT HV\frac{d v}{d t} \in T^H V for all tt; with the initial condition s(b)=v(0)s(b) = v(0).

General properties and axioms

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

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

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

X(fs)=(Xf)s+f X(s),fC (M),sΓξ. \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 ξ=(EM)\xi = (E\to M), a covariant differentiation is any R\mathbf{R}-linear map :ΓξΓ(T *Mξ)\nabla:\Gamma \xi\to \Gamma (T^* M\otimes \xi) such that (fs)=dfs+fs\nabla(f s) = df\otimes s + f\nabla s where fC (M)f\in C^\infty(M) and sΓξs\in \Gamma \xi. (We sometimes write sloppy ΓE\Gamma E meaning Γξ\Gamma\xi).

It appears that such differentiations are in 1-1 correspondence with covariant derivatives coming from connections. Namely, let e 1,,e ne_1,\ldots,e_n be a basis of Γ Uξ\Gamma_U \xi above some contractible neighborhood UMU\subset M. Then e j= i,kΓ kj idx ke i\nabla e_j = \sum_{i,k} \Gamma^i_{k j} d x^k\otimes e_i for some functions Γ kj i=Γ kj i(x)\Gamma^i_{k j} = \Gamma^i_{k j}(x) on MM. It appears that these functions, under the coordinate changes, change as the Christoffel symbols for some connection. One may also write e j= iω j ie j\nabla e_j = \sum_i \omega^i_j \otimes e_j. Then for the arbitrary section s= is ie iΓ Uξs = \sum_i s^i e_i \in \Gamma_U \xi we have (by R\mathbf{R}-linearity and Leibniz rule) s= i(ds i+ω j is j)e i\nabla s = \sum_i (d s^i + \omega^i_j s^j)\otimes e_i.

Parallel transport, holonomy and monodromy

Let π:EM\pi:E\to M be a principal GG-bundle with an Ehresmann connection and u:I=[0,t 0]Mu: I = [0,t_0]\to M a regular smooth curve with u(0)=xu(0) = x and u(t 0)=yu(t_0) = y. Then for each eπ 1(e)e\in \pi^{-1}(e), there is a unique horizontal lift of uu, i.e. a smooth curve v:IEv:I\to E such that πv=u\pi\circ v = u and at each point the tangent vector to vv is horizontal, i.e. belongs to the horizontal subspace T v(0) HET v(0)ET^H_{v(0)} E\subset T_{v(0)} E. Define Π u(e)=v(t 0)π 1(y)\Pi_u(e)=v(t_0)\in \pi^{-1}(y). The rule Π u:E xE y:eΠ u(e)\Pi_u :E_x\to E_y : e\mapsto \Pi_u(e) is the parallel transport along curve uu.

By the equivariance of the field of horizontal subspaces, it follows that if x=yx=y, i.e. uu is a smooth loop, Π u\Pi_u is the right shift by a unique element gGg\in G, hence Π u\Pi_u can be considered an element in GG. The holonomy subgroup at point xx is the group of all Π uG\Pi_u\in G for all smooth loops uu at xx. It is a Lie subgroup of GG. 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 GG.

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 :ΓEΓ(T *ME)ΓT *M C (M)ΓEΩ 1(M) C (M)ΓE\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 R\mathbf{R}-linear map ˜:ΓEΩ MΓEΩ +1M\tilde\nabla : \Gamma E\otimes \Omega^\bullet M \to \Gamma E\otimes \Omega^{\bullet+1} M, such that ˜| ΓE=\tilde\nabla|_{\Gamma E} = \nabla and such that for all sΓEs\in \Gamma E and homogeneous ωΩ (M)\omega\in\Omega^\bullet(M), the Leibniz rule holds

˜(ωs)=dωs+(1) degωω(s). \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 Ω=˜:ΓEΓ( 2T *ME)=Ω 2(M)ΓE\Omega = \tilde{\nabla}\circ\nabla : \Gamma E\to \Gamma(\wedge^2 T^* M\otimes E) = \Omega^2(M)\otimes\Gamma E.

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

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

(˜)(fs) = ˜(dfs+f(s)) = ddfsdf(s)+df(s)+f˜((s)) = f(˜)(s)\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 1MF_1\to M, F 2MF_2\to M dva glatka vektorska svežnja i ΓF 1\Gamma F_1 i ΓF 2\Gamma F_2 prostori glatkih prereza tih svežnjeva nad parakompaktnom glatkom mnogostrukošću MM, koje možemo promatrati kao C (M)C^\infty(M)-module. Neka je D:ΓF 1ΓF 2D:\Gamma F_1\to \Gamma F_2 proizvoljno preslikavanje koje ima jako svojstvo da je C (M)C^\infty(M)-linearno (ne samo R\mathbf{R}-linearno). Dokaži da postoji dobro definirano preslikavanje D 0:F 1F 2D_0 : F_1\to F_2 vektorskih svežnjeva nad MM takvo da za svaki prerez sΓF 1s\in\Gamma F_1 i za svaku točku xMx\in M vrijedi D(s)(x)=D 0(s(x))D(s)(x) = D_0(s(x)).

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

Re j= iΩ j ie i R\circ e_j = \sum_i \Omega^i_j \otimes e_i

Then

Ω j i=dω j i+ω p iω j p \Omega^i_j = d \omega^i_j + \omega^i_p\wedge \omega^p_j

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

Ambrose-Singer’s theorem

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

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