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suspected the terrible truth but learnt about it only in 1945: a deadly blow from which he never recovered. Henri Cartan followed his father steps and became a very prominent mathematician.
3.3Connections on Principal Bundles: The Mathematical Definition
After presenting the previous historical introduction we come to the contemporary mathematical definition of a connection on a principle fibre-bundle. The adopted view-point is that of Ehresmann and what we are going to define is usually named an Ehresmann connection in the mathematical literature.
To this effect we need a number of mathematical preliminaries on Lie groups which will be essential not only in the forthcoming discussion of a principal connection but will play a very important conceptual role also in all subsequent developments of the theory and in each of its applications.
3.3.1 Mathematical Preliminaries on Lie Groups
Firstly we recall that on a Lie group G one can define two transitive actions of the same group on itself, the left and the right multiplication respectively. Indeed to each element γ G we can associate two continuous, infinitely differentiable and invertible maps of G in G, named the left and the right translation which we introduced in (2.4.2). Secondly we introduce the concepts of pull-back and pushforward of any diffeomorphism mapping a differential manifold M into an open submanifold of another differentiable manifold N . Let φ be any such map
φ : M → N |
(3.3.1) |
the push-forward of φ, denoted φ , is a map from the space of sections of the tangent bundle T M to the space of sections of the tangent bundle T N :
φ : Γ (T M , M ) → Γ (T N , N ) |
(3.3.2) |
Explicitly if X Γ (T M , M ) is a vector field over M , we can use it to define a new vector field φ X Γ (T N , N ) over N using the following procedure. For any f C∞(N ), namely for any smooth function on N , the action of φ X on such a function is given by:
φ X(f ) ≡ X(f ◦ φ) ◦ φ−1 |
(3.3.3) |
To clarify this concept let us describe the push-forward in a pair of open charts for both manifolds. Consider Fig. 3.11. On the open neighborhood U M we have
3.3 Connections on Principal Bundles: The Mathematical Definition |
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Fig. 3.11 Graphical description of the concept of push-forward
coordinates xμ while on the open neighborhood V N we have coordinates yν . In this pair of local charts {U, V } the diffeomorphism (3.3.1) is described by giving the coordinates yν as smooth functions of the coordinates xμ:
yν = yν (x) ≡ φν (x) |
(3.3.4) |
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On the neighborhood V the smooth function f |
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function f (y) of the n-coordinates y . It follows that f ◦ φ is a smooth function f˜(x) on the open neighborhood U M simply given by:
f˜(x) = f y(x) |
(3.3.5) |
Any tangent vector X Γ (T M , M ) is locally described on the open chart U by a first order differential operator of the form:
∂ |
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X = Xμ(x) ∂xμ |
(3.3.6) |
which therefore can act on f˜(x):
Xf˜(x) = Xμ(x) |
∂yν ∂ |
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Considering now the coordinates xμ on U as functions of the inverse of the diffeomorphism φ:
xμ = xμ(y) = φ−1(y) μ
(3.3.7)
yν on V , through the
(3.3.8)
we realize that (3.3.7) defines a new linear first order differential operator acting on any function f : V → R. This differential operator is the push-forward of X through the diffeomorphism φ:
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(3.3.9) |
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Similarly the pull-back of φ, denoted φ , is a map from the space of sections of the cotangent bundle T N to the space of sections of the cotangent bundle T M :
φ : Γ T N , N → Γ T M , M |
(3.3.10) |
Explicitly if ω Γ (T N , N ) is a differential one-form over N , we can use it to define a differential one-form φ ω over M as it follows. We recall that a one-form is defined if we assign its value on any vector-field field, hence we set:
X Γ (T M , M ); φ ω(X) ≡ ω(φ X) |
(3.3.11) |
Considering (3.3.9), the local description of the pull-back on a pair of open charts {U, V } is easily derived from the definition (3.3.11). If we name ωμ(y) the local components of the one-form ω on the coordinate patch V :
ω = ωμ(y) dyμ |
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the components of the pull-back are immediately deduced: |
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φ ω = φ ω μ(x) dxμ ≡ ων y(x) |
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(3.3.13) |
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3.3.1.1 Left-/Right-Invariant Vector Fields
Let us now consider the case where the manifold M coincides with manifold N and both are equal to a Lie group manifold G. The left and the right translations defined in (2.4.2) are diffeomorphisms and for each of them we can consider both the push-forward and the pull-back. This construction allows to introduce the notion of left- (respectively right-) invariant vector fields and one-forms over the Lie group manifolds G.
Definition 3.3.1 A vector field X Γ (T G, G) defined over a Lie group manifold G is named left-invariant (respectively right-invariant) if the following condition holds true:
γ G : Lγ X = X (respectively Rγ X = X) |
(3.3.14) |
Similarly:
Definition 3.3.2 A one-form σ Γ (T G, G) defined over a Lie group manifold G is named left-invariant (respectively right-invariant) if the following condition holds true:
γ G : Lγ σ = σ |
respectively Rγ σ = σ |
(3.3.15) |
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Let us recall that the space of sections of the tangent bundle has the structure of an infinite dimensional Lie algebra for any manifold M . Indeed, given any two sections, we can compute their commutator as differential operators and this defines the necessary Lie bracket:
X, Y Γ (T M , M ); [X, Y] = Z Γ (T M , M ) |
(3.3.16) |
Viewed as a Lie algebra the space of sections of the tangent bundle is usually denoted Diff0(M ) since every vector field can be regarded as the generator of a diffeomorphism infinitesimally close to the identity.
In the case of group manifolds we have the following simple but very fundamental theorem:
Theorem 3.3.1 The two sets of left-invariant and of right-invariant vector fields over a Lie group manifold G close two finite Lie subalgebras of Diff0(G), respectively named GL/R , which are isomorphic to each other and to the abstract Lie algebra G of the Lie group G. Furthermore GL/R commute with each other.
The proof of this theorem is obtained through a series of steps and through the proof of some intermediate lemmas. Let us begin with the first.
Proof For any diffeomorphism φ the push-forward map φ has the following property:
X, Y Diff0(M ) : φ [X, Y] = [φ X, φ Y] |
(3.3.17) |
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No proof is required for this lemma since it follows straightforwardly from the definition (3.3.3) of the pushforward. However the consequences of the lemma are far reaching. Indeed it implies that the commutator of two left-invariant (respectively right-invariant) vector fields is still left-invariant (respectively right-invariant). So we have the
Lemma 3.3.1 The two sets GL/R of left-invariant (respectively right-invariant) vector fields constitute two Lie subalgebras GL/R Diff0(G).
X, Y GL/R : [X, Y] GL/R |
(3.3.18) |
This being established we can now show that the left-invariant vector fields can be put into one-to-one correspondence with the elements of tangent space to the group manifold at the identity element, namely with TeG. From this correspondence it will follow that the Lie subalgebra GL has dimension equal to the dimension n of the Lie group. The same correspondence can be established also for the right-invariant vector fields and the same conclusion about the dimension of the Lie algebra GR can be deduced. The argument goes as follows.
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Let g : [0, 1] → G be a path in G with initial point at the identity element g(0) = e G. In local coordinates on the group manifold the path g is described as follows:
t [0, 1]; G g(t) = g α1(t) . . . αn(t) |
(3.3.19) |
where αi denote the group parameters and g(α) denotes the by the parameters α. Hence we obtain:
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group element identified
(3.3.20)
(3.3.21)
constitutes the component of a tangent vector at the identity that we name Xe :
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(3.3.22) |
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Conversely for each Xe TeG we can construct a path g(t) that admits Xe as tangent vector at the identity element. Given a tangent vector t at a point p M of a manifold, constructing a curve on M which goes through p and admits t as tangent vector in that point is a problem which admits solutions on any differentiable manifold, a fortiori on a Lie group manifold G. Let therefore g(t) be such a path. To Xe we can associate a left-invariant vector field XL defining the action of the latter on any smooth function f as it follows:
f C∞(G) and ρ G : XLf (ρ) ≡ f |
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(3.3.23) |
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Applying the definition in (3.3.3) the reader can immediately verify that XL defined above is left-invariant. In a completely analogous way, to the same tangent vector Xe we can associate a right-invariant vector field XR , setting:
f C∞(G) and ρ G : XR f (ρ) ≡ f |
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(3.3.24) |
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In this way we have established that to each tangent vector at the identity we can associate both a left-invariant and a right-invariant vector field. On the other hand since each vector field X is a section of the tangent bundle, namely a map:
p M : p → Xp TpM |
(3.3.25) |
it follows that each left-invariant or right-invariant vector fields singles out a tangent vector at the identity. The relevant point is that this double correspondence is an isomorphism of vector spaces. Indeed we have:
Lemma 3.3.2 Let G be a Lie group, let GL/R be the Lie algebra of left-invariant (respectively right-invariant) vector fields. The correspondence:
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X GL/R π : X → Xe TeG |
(3.3.26) |
is an isomorphism of vector spaces. |
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Proof Let us first of all observe that the correspondence (3.3.26) is a linear map. Indeed for a, b R and X, Y GL/R we have:
π(aX + bY) = aXe + bYe |
(3.3.27) |
In order to show that π is an isomorphism we need to prove that π is both injective and surjective, in other words that:
ker π = 0; Im π = TeG |
(3.3.28) |
The second of the two conditions (3.3.28) was already proved. Indeed we have shown above that to each tangent vector Xe at the identity we can associate a leftinvariant (right-invariant) vector field which reduces to that vector at g = e. It remains to be shown that ker π = 0. This follows from a simple observation. If X is left-invariant, then its value at the group element γ G is determined by its value at the identity by means of a left translation, namely:
Xγ = Lγ Xe |
(3.3.29) |
Therefore if Xe = 0 it follows that Xγ = 0 for all group elements γ G and hence X = 0 as a vector field. Hence there are no non-trivial vectors in the kernel of the map π and this concludes the proof of the lemma.
We have therefore:
Corollary 3.3.1 The dimensions of the two Lie algebras GL and GR are equal among themselves and equal to those of the abstract Lie algebra G
dim GL = dim GR = dim G = dim TeG |
(3.3.30) |
In this way we have shown the isomorphism GL GR G as vector spaces. It remains to be shown the same isomorphism as Lie algebras.
Proof We can now complete the proof of Theorem 3.3.1. To this effect we argue in the following way. Utilizing the just established vector space isomorphism let us choose a basis for TeG which we denote as tA (A = 1, . . . , dim G). In this way we have:
t TeG : t = xAtA, xA R |
(3.3.31) |
Relying on the constructions (3.3.23) and (3.3.24) to each element of the basis {tA} we can associate the corresponding left- (right-)invariant vector field:
TA(L) |
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where gA(t) denotes a path in G passing through the identity and there admitting the vector tA as tangent. The established vector space isomorphism guarantees that T(L)A and T(L)A constitute a basis respectively for GL and GR . Hence we can write:
TA(L), TB(L) = CAB(L)C TC(L) |
(3.3.34) |
TA(R), TB(R) = CAB(R)C TC(R) |
(3.3.35) |
where CAB(L)C and CAB(L)C are constants that, a priori, might be completely different. Equations follow from the fact that we have established that both GL and GR are dimension n Lie algebras and the generators T(L/R)A constitute a basis.
Let us know calculate explicitly the commutators of the basis elements using
their definitions. On any function f : G → R, we find: |
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The above (3.3.37) hold true at any point ρ G. Evaluating them at the identity ρ = e and taking their sum we obtain:
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(3.3.38) |
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which implies: |
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CAB(L)C = −CAB(R)C |
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This relation suffices to establish the Lie algebra isomorphism of GL with GR . Indeed they are isomorphic as vector spaces and their structure constants become identical under the very simple change of basis:
TA(L) ↔ −TA(R) |
(3.3.40) |
This concludes the proof of Theorem 3.3.1. Indeed the last isomorphism advocated in that proposition amounts simply to a definition.
Definition 3.3.3 The Lie algebra GL of the left-invariant vector fields on the Lie group manifold G, isomorphic to that of the right-invariant ones GR , is named the Lie algebra G of the considered Lie group.