A Classification Theorem for Connections

Kostake Teleman

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract

In this paper we put together ideas and results related to the geometric theory of connections, with the hope that, on such bases, the applications to Physics will become a little bit more conceptual.

Mathematics Subject Classification: 53C05 Key words: bundle, connection holonomy, acyclic covering, homology group

1 Introduction

The theory of connections arose in Riemannian geometry by the grace of Levi Civita and became soon a part of Physics in the works of A.Einstein, H.Weyl, E.Cartan, C.T.Yang and Mills. The geometrical spirit dominated the works o these giants and of their followers, until the hightech penetrated the field, leading to important achievements, especially in Differential Topology. These trends shadowed the geometrical intuition, but recent work, especially due to E.Witten, brought the geometrical thinking into Physics once more.

It is the first goal of this Note to remind the origins of the geometrical theory of connections, which can be found in the work of H.Poincaré, who invented covering spaces and the fundamental group.

The second goal is to generalize Poincaré’s construction of universal coverings and fundamental groups. As a result, we produce the universal principal bundle associated with a connected manifold. This principal bundle is named universal because every differentiable finite dimensional bundle with structure group is associated to the universal bundle.

2 Connections on complex vector line bundles

Let M be an orientable, connected, compact manifold and consider a complex vector line bundle L = (E, p, M) endowed with a connection C. Denote by Ω ∈ ∧^{2}(M) the curvature form of the connection C. Since Ω is closed, for each contractible neighbourhood U ⊂ M there exists a 1–form φ_{U} ∈ ∧^{1}(U) such that

Balkan Journal of Geometry and Its Applications, Vol.5, No.1, 2000, pp. 157-162 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Ω|_{U} = dφ_{U}.

When U is an pen covering of M consisting in contractible sets with contractible double intersections U ∩ V, U ∈ U, V ∈ U, we can find a system of 1–forms {φ_{U}}_{U∈U} and a system of functions F_{UV} : U ∩ B → C such that

Ω = dφ_{U} on U, φ_{U} − φ_{V} = df_{UV} on U ∩ U.

Let (h_{1}, …, h_{m}, k_{1}, …, k_{r}) be a system of generators of the homology group H_{1}(M, Z) and denote by x_{1}, …, x_{m}, y_{1}, … y_{r} smooth closed paths representing the homology classes h_{j}, k_{s}. We can suppose that all linear relations between x_{j}, y_{s} are consequences of torsion relations

p_{s}k_{s} = 0, s = 1, …, r, p_{s} ∈ N.

Denote by c_{1} = ρ(x_{1}), …, c_{m} = ρ(x_{m}), d_{1} = ρ(y_{1}), …, d_{r} = ρ(y_{r}) the (non vanishing) complex numbers representing the holonomy automorphisms corresponding to the paths x_{j}, y_{s}. According to one of the famous de Rham theorems, there exists a closed complex–valued 1–form α such that

exp (∫_{x_{j}} α) = c_{j}, j = 1, …, m.

Then we have this almost obvious Proposition. The bundle L and the connection C determined, up to isomorphisms, by the system

(Ω, α, x, y, d),

where

x = (x_{1}, …, x_{m}), y = (y_{1}, …, y_{r}), d = (d_{1}, …, d_{r}).

The numbers (d_{j})^{p_{j}} are known when Ω and y are given. Proof. We make use of the fact that the holonomy group of the connection C is abelian and that, for each closed path x in M there exist integers z_{1}, …, z_{m}, u_{1}, …, u_{r} and 2–chains σ, σ_{1}, …, σ_{r} such that

x = z_{1}x_{1} + … + z_{m}x_{m} = u_{1}y_{1} + … + u_{r}y_{r} + dσ, p_{s}y_{s} = dσ_{s}.

Then we shall have, by using Stokes’ theorem,

(d_{1})^{p_{1}} = exp (∫_{σ_{1}} Ω) , …, (d_{r})^{p_{r}} = exp (∫_{σ_{r}} Ω)

and the holonomy automorphism ρ(x) will be represented by the complex number:

c(x) = exp (∫_{σ} Ω) (c_{1})^{z_{1}} … (c_{m})^{z_{m}}(d_{1})^{u_{1}} … (d_{r})^{u_{r}}.

The Proposition is now a direct consequence of the following general Theorem, which will be proved in the next sections:

Theorem I . A pair (E, C), consisting in a differentiable vector bundle E and in a linear connection C in this bundle, is determined, up to isomorphisms, by the holonomy representation of the connection. Corollary 1. When the group H_{1}(M, Z) is torsion-free, the isomorphism class [E, C] of the pair (E, C) is determined by the system (Ω, α, x). Corollary 2. When the manifold M is simply connected, the isomorphism class [E, C] is determined by the curvature form Ω alone. Remark. For each integer 2-cycle Z of M, the period ∫ ZΩ is an integer.

3 A generalized fundamental group

Let M be a connected differentiable manifold and let a, b be points in M. We denote by P (a, b) the set of continuous paths of M with endpoints a, b and which are smooth excepting finite sets of points. Then introduce an equivalence relation in the set P (a, b) by considering equivalent two paths c_{1}, c_{2} which define the same holonomy isomorphism h : E_{a} → E_{b} for all connexions C in all bundles E → M. Let P′(a, b) the set of equivalence classes. There is a natural composition law for paths P (a, b) × P (b, q) → P (a, q), which induces a multiplication P′(a, b) × P′(b, q) → P′(a, q). Let Q(a), Q′(a), G(a), G′(a) be the sets

Q(a) = ⋃{P (a, b); b ∈ M}, Q′(a) = ⋃{P′(a, b); b ∈ M}
G(a) = P (a, a), G′(a) = P′(a, a).

Then it is a question of routine to prove the following statements, by repeating the guidelines leading to Poincaré’s universal coverings:

1. G′(a) is a group and the multiplication of equivalence classes induces a left action

G′(a) × Q′(a) → Q′(a).

2. When (E, C) is a bundle–connection pair, the holonomy construction provides natural maps

ρ(C, a) : G′(a) → G(E_{a}), λ(C, a) : E_{a} × Q′(a) → E,

where G(E_{a}) is the group of automorphisms of the fibre E_{a}.

3. The map ρ(C, a) is a morphism of groups and is related to λ(C, a) in a way which is expressed by the equality:

λ(C, a)(e, ω α) = (ρ(C, a)(ω))(λ(C, a)(e, α))
e ∈ E_{a}, ω ∈ G′(a), α ∈ Q′(a).

4. The group G′(a) is canonically embeded as a dense subgroup of the projective limit of the groups G(E_{a}) and Q′(a) is canonically embeded as a dense subset of the projective limit of the bundles E → M. These embedings induce topologies on G′(a) and Q′(a) such that G′(a) becomes a topological group acting continously on Q′(a).

5. The map

λ_{C,a} : E_{a} × Q′(a) → E

is onto and E arises as coset space of E_{a} × Q′(a) and of the equivalence relation given by

(e, ω α)(ρ(C, a)(ω)(e), α).

6. There is a canonical projection p_{a} : Q′(a) → M defined as follows:

p_{a}(α) = b if α ∈ P′(a, b)

and the triple π(M, a) = (Q′(a), p_{a}, M) is a topological, locally trivial, principal bundle with structure group G′(a).

7. Every differentiable bundle ξ over M, with structure group G, is associated with the principal bundle π(M, a) and with a continuous homomorphism ρ : G′(a) → G. Such a homomorphism defines a connection on the bundle ξ.

Theorem I is a direct consequence of these properties, from the topological background view. In order to recover the differentiable structures of the bundle–connection pairs, we have to introduce differentiable structures in the triple π(M, a). Definition. The bundle π(M, a) will be named the universal bundle of the pair (M, a) and the group G′(a) will be named the generalized fundamental group of (M, a).

4 Generalized connections

Let ξ = (E, p, M) be a differentiable bundle with fibres E_{b} = p^{−1}(b).

We denote by C(M) the category whose objects are points of M and whose sets of morphisms are the sets P′(a, b). And we denote by C(ξ) the category whose objects are the fibres E_{b} and whose morphisms are the diffeomorphisms f : E_{a} → E_{b}.

When ξ is a bundle with structure group G, each fibre E_{a} is endowed with a group G_{a} of automorphisms of E_{a}, which is isomorphic to the group G.

When ξ is a bundle with structure group G, we denote by C(ξ, s) the subcategory of C(ξ) having the same objects as Cξ) and whose morphisms f : E_{a} → E_{b} are the diffeomorphisms with the property

g ∈ G_{a} ⇒ f gf^{−1} ∈ G_{b}.

The categories C(ξ), C(ξ, s) have canonical differentiable structures.

Suppose C is a given connection on ξ. Then the holonomy construction provides differentiable morphisms of categories

F_{C} : C(M) → C(ξ), C(ξ, s).

Conversely, suppose F : C(M) → C(ξ, s) is a given differentiable morphism. In this case, by restriction, we get a homomorphism of groups

ρ′_{G}(a) → G_{a}

and an associated bundle η with a connection C. The bundle η is canonically isomorphic to ξ and the canonic isomorphism φ : η → ξ allows us to transport the connection C and get a connection C′ on ξ. We proved: Theorem II. Each differentiable morphism of categories

F : C(M) → C(ξ, s)

defines a connection on ξ.

5 Subgroups of G′(m) associated with acyclic open coverings

Let H be a set of pairs (U, h_{U}) the first terms of which form an open acyclic covering U of M and such that each h_{U} is a diffeomorphism from U to the open unit ball B = {x ∈ br^{n} : |x| < 1}, where n is the dimension of M.

Let further m be a fixed point in M. For each U ∈ U, we define m_{U} = (h_{U})^{−1}(0) and select a path λ_{U} with end points m, m_{U}. Then for each point x ∈ U we denote x_{U} = h_{U}(x). For a ∈ B we denote by c_{a} : [0, 1] → B the straight path in B with endpoints 0, a. When a = x_{U}, we denote the path c_{a} by C_{x,U} and consider the path l_{x,U} = (h_{U})^{−1}x_{x,U}.

Finally, we suppose that, for each couple (x, y) of points in M, which belong at least to a set U ∈ U, we selected a path m_{x,y}, with endpoints x, y, depending differentiably on x and y, but not depending on U. This is always possible, using for instance a Riemannian structure on M and minimal geodesic arcs.

The paths

λ_{U} ∈ P (m, m_{U}), l_{x,U} ∈ P (m_{U}, x), m_{x,y} ∈ P (x, y)

define equivalence classes

[λ_{U}] ∈ P′(m, m_{U}), [l_{x,U}] ∈ P′(m_{U}, x), [m_{x,y}] ∈ P′(x, y).

We define further:

for x ∈ U, y ∈ U, ω_{U,x,y} = [λ_{U}][l_{x,U}][l_{y,U}]^{−1}[λ_{U}]^{−1} ∈ G′(m)

for x ∈ U ∩ V, ω_{U,V,x} = [λ_{U}][l_{x,U}][l_{x,V}]^{−1}[λ_{V}]^{−1}

for x ∈ U, y ∈ U, z ∈ U, ω_{U,x,y,z} = ω_{U,x,y}ω_{U,y,z}ω_{U,z,x} ∈ G′(m).

Then we have the identities:

ω_{U,V,y} = ω_{U,y,x}ω_{U,V,x}ω_{V,x,y}

ω_{U,x,y} = ω_{U,y,x}
ω_{U,V,x}ω_{v,W,x} = ω_{U,W,x}.

For each non empty double or triple intersection, let us select points

x_{U,V} ∈ U ∩ V, x_{U,V,W} ∈ U ∩ V ∩ W

not depending on the order of the intersected sets. Denote

ω_{U,V} = ω_{U,V,x_{U,V}} , ω_{U,V,W} = ω_{U,x_{U,V} ,x_{U,V,W}} .

Then, for x ∈ U ∩ V, we have ω_{U,V,x} = ω_{x,x_{U,V}} ω_{U,V} ω_{V,x_{U,V,x}} and, in particular, ω_{U,V,x_{U,V,W}} = (ω_{U,V,W})^{−1}ω_{U,V} ω_{V,U,W}. Let us denote

p_{U,V,W} = (ω_{U,V,W})^{−1}ω_{U,V} ω_{V,U,W} .

Then

p_{U,V,W} p_{V,W,U}p_{W,U,V} = 1.

Let us denote by G(m, U) the subgroup of G′(m) generated by the elements ω_{U,x,y}, ω_{U,V,x}; the group G(m, U) carries a differentiable structure consisting in the family of maps f_{U} : U × U → G(m, U), f_{U}(x, y) = ω_{U,x,y}, f_{UV} : U ∩ V → G(m, U), f_{UV} (x) = ω_{U,V,x}. The group G(m, U) is generated by the elements ω_{U,x,y} and ω_{UV}.

6 Computation of the coefficients of a connection

Suppose it is given a linear connection in a vector bundle ξ = (E, p, M) with fibre F = E_{m}. Then we have a homomorphism of groups ρ : G′(a) → GL(F) and we can consider the maps:

g_{U} : U × U → GL(F), g_{U}(x, y) = ρ(ω_{U,x,y})
g_{U,V} : U ⋂ V → GL(F), g_{U,V} (x) = ρ(ω_{U,V,x}).

Using the holonomy along the paths λ_{U}, we can identify canonically each set p^{−1}(U), with U × F and, considering the charts h_{U}, u ∈ U, the connection form A_{i}dx^{i} will be defined, on each U, by the relation

A_{i}(x) = \frac{∂g_{U}(x, y)}{∂y^{i}}|_{(x,x)}.

The maps g_{U}, g_{U,V}, enjoy some properties that are consequences of the relation verified by the ω’s. Theorem II. Let F be a real vector space of finite dimension. Then with each differentiable vector bundle-connection pair (E, C), there exists a family of differentiable maps

G = (g_{U} : U × U → GL(F), g_{UV} : U ⋂ V → GL(F))

subject to the relations

\begin{matrix}g_{UV} (y) = g_{U}(y, x)g_{UV} g_{V} (x, y) \\ g_{UV} (x)g_{V W} (x) = g_{V W} (x). \end{matrix}

Conversely, given a family G enjoying the properties above, there exists a unique class of isomorphism of bundle-connection pairs (E, C) such that G is associated with (E, C).

When we want to compare this theorem with the Proposition given in Section 1, we recognize that the role of Ω is played by the system {g_{U}}, while the role of α is played by the system {g_{U,V} }.

References

[1] S.Kobayashi and K.Nomizu, Foundations of Differential Geometry, (1963), Interscience Publishers, New-York, London.

University of Bucharest Faculty of Mathematics Str. Academiei no 14 70109, Bucharest, Romania