118 |
3 Connections and Metrics |
Since the forms σ are Lie algebra-valued, they can be expanded along a basis of generators tA for G, as we already did in (3.3.49) and the same can be done for the curvature two-form F[σ ], namely we get:
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F[σ ] = FA[σ ]tA |
(3.3.60) |
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FA[σ ] = dσ A + |
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CBC A σ B σ C |
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where C |
A are the structure constants of the Lie algebra: |
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BC |
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[tB , tC ] = CBC A tA |
(3.3.61) |
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and the Maurer Cartan equations (3.3.59) amount to the statement: |
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FA[σL/R ] = 0 |
(3.3.62) |
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3.3.2 Ehresmann Connections on a Principle Fibre Bundle
Equipped with the notions developed in the previous Sect. we are finally in a position to introduce the rigorous mathematical definition due to Ehresmann (see Fig. 3.12) of a connection on a generic principle bundle.
Let P (M , G) be a principle fibre-bundle with base-manifold M and structural group G. Let us moreover denote π the projection:
π : P → M |
(3.3.63) |
Consider the action of the Lie group G on the total space P :
g G g : G → G |
(3.3.64) |
By definition this action is vertical in the sense that
u P , g G : π g(u) = π(u) |
(3.3.65) |
namely it moves points only along the fibres. Given any element have denoted by G the Lie algebra of the structural group, we one-dimensional subgroup generated by it
X G where we can consider the
gX (t) = exp[tX], t R |
(3.3.66) |
and consider the curve CX(t, u) in the manifold P obtained by acting with gX (t) on some point u P :
CX(t, u) ≡ gX (t)(u) |
(3.3.67) |
3.3 Connections on Principal Bundles: The Mathematical Definition |
119 |
Fig. 3.12 Charles Ehresmann was born in German speaking Alsace in 1905 from a poor family. His first education was in German, but after Alsace was returned to France in 1918 as a result of Germany’s defeat in World War I, Ehresmann attended only French schools and his University education was entirely French. Indeed in 1924 he entered the École Normale Superiéure from which he graduated in 1927. After that, he served as teacher of Mathematics in the French colony of Morocco and then went to Göttingen that in the late twenties and beginning of the thirties was the major scientific center of the world, at least for Mathematics and Physics. The raising of Nazi power in Germany dismantled the scientific leadership of the country, caused the decay of Göttingen and obliged all the Jewish scientists who so greatly contributed to German culture to emigrate to the United States. Ehresmann also fled from Göttingen to Princeton where he studied for few years until 1934. In that year he returned to France to obtain his doctorate under the supervision of one of the giants of mathematical thought of the XXth century: Elie Cartan. Charles Ehresmann was professor at the Universities of Strasbourg and Clermont Ferrand. In 1955 a special chair of Topology was created for him at the University of Paris which he occupied up to his retirement in 1975. He died in 1979 in Amiens where his second wife, also a mathematician held a chair. Charles Ehresmann was one of the creators of differential topology. He greatly contributed to the development of the notion of fibre-bundles and of the connections defined over them. He founded the mathematical theory of categories
The vertical action of the structural group implies that: |
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π CX (t, u) = p M if π(u) = p |
(3.3.68) |
These items allow us to construct a linear map # which associates a vector field X# over P to every element X of the Lie algebra G:
# : G X → X# Γ (T P , P ) |
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(3.3.69) |
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d |
CX(t, u)%t |
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0 |
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f C∞(P ) X#f (u) ≡ dt f |
= |
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% |
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Focusing on any point u P of the bundle, the map (3.3.69) reduces to a map:
#u : G → TuP |
(3.3.70) |
120 |
3 Connections and Metrics |
Fig. 3.13 The tangent space to a principle bundle P splits at any point u P into a vertical subspace along the fibres and a horizontal subspace parallel to the base manifold. This splitting is the intrinsic geometric meaning of a connection
from the Lie Algebra to the tangent space at u. The map #u is not surjective. We introduce the following:
Definition 3.3.4 At any point p P (M, G) of a principle fibre bundle we name vertical subspace VuP of the tangent space TuP the image of the map #u:
TuP Vu ≡ Im #u |
(3.3.71) |
Indeed any vector t Im #u lies in the tangent space to the fibre Gp , where p ≡ π(u).
The meaning of Definition 3.3.4 becomes clearer if we consider both Fig. 3.13 and a local trivialization of the bundle including u P . In such a local trivialization
the bundle point u is identified by a pair: |
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u loc. triv. (p, f ) where |
p M |
(3.3.72) |
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−→ |
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f |
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G |
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and the curve CX (t, u) takes the following appearance: |
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loc. triv. |
p, etXf |
, |
p = π(u) |
(3.3.73) |
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CX (t, u) −→ |
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Correspondingly, naming αi the group parameters, just as in the previous section, and xμ the coordinates on the base space M , a generic function f C∞(P ) is just a function f (x, α) of the m coordinates x and of the n parameters α. Comparing (3.3.69) with (3.3.24) we see that, in the local trivialization, the vertical tangent vector X# reduces to nothing else but to the right-invariant vector field on the group manifold associated with the same Lie algebra element X:
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loc. triv. |
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∂ |
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X# |
−→ |
XR = XRi (α) |
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(3.3.74) |
∂αi |
As we see from (3.3.74), in a local trivialization a vertical vector contains no derivatives with respect to the base space coordinates. This means that its projection onto the base manifold is zero. Indeed it follows from Definition 3.3.4 that: