Connection form

A linear differential form $ \theta $ on a principal fibre bundle $ P $ that takes values in the Lie algebra $ \mathfrak g $ of the structure group $ G $ of $ P $. It is defined by a certain linear connection $ \Gamma $ on $ P $, and it determines this connection uniquely. The values of the connection form $ \theta _ {y} ( Y) $ in terms of $ \Gamma $, where $ y \in P $ and $ Y \in T _ {y} ( P) $, are defined as the elements of $ \mathfrak g $ which, under the action of $ G $ on $ P $, generate the second component of $ Y $ relative to the direct sum $ T _ {y} ( F) = \Delta _ {y} \oplus T _ {y} ( G _ {y} ) $. Here $ G _ {y} $ is the fibre of $ P $ that contains $ y $ and $ \Delta $ is the horizontal distribution of $ \Gamma $. The horizontal distribution $ \Delta $, and so the connection $ \Gamma $, can be recovered from the connection form $ \theta $ in the following way.

The Cartan–Laptev theorem. For a form $ \theta $ on $ P $ with values in $ \mathfrak g $ to be a connection form it is necessary and sufficient that: 1) $ \theta _ {y} ( Y) $, for $ Y \in T _ {y} ( G _ {y} ) $, is the element of $ \mathfrak g $ that generates $ Y $ under the action of $ G $ on $ P $; and 2) the $ \mathfrak g $-valued $ 2 $-form

$$ \Omega = d \theta + \frac{1}{2} [ \theta , \theta ] , $$

formed from $ \theta $, is semi-basic, or horizontal, that is, $ \Omega _ {y} ( Y , Y _ {1} ) = 0 $ if at least one of the vectors $ Y , Y _ {1} $ belongs to $ T _ {y} ( G _ {y} ) $. The $ 2 $-form $ \Omega $ is called the curvature form of the connection. If a basis $ \{ e _ {1}, \dots, e _ {r} \} $ is defined in $ \mathfrak g $, then condition 2) can locally be expressed by the equalities:

$$ d \theta ^ \rho + \frac{1}{2} C _ {\sigma \tau } ^ \rho \theta ^ \sigma \wedge \theta ^ \tau = \ \frac{1}{2} R _ {ij} ^ \rho \omega ^ {i} \wedge \omega ^ {j} , $$

where $ \omega ^ {1}, \dots, \omega ^ {n} $ are certain linearly independent semi-basic $ 1 $-forms. The necessity of condition 2) was established in this form by E. Cartan [1]; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev [2]. The equations

for the components of the connection form are called the structure equations for the connection in $ P $, the $ R _ {ij} ^ \rho $ define the curvature object.

As an example, let $ P $ be the space of affine frames in the tangent bundle of an $ n $-dimensional smooth manifold $ M $. Then $ G $ and $ \mathfrak g $ are, respectively, the group and the Lie algebra of matrices of the form

$$ \left \| \begin{array}{cc} 1 &a ^ {i} \\ 0 &A _ {j} ^ {i} \\ \end{array} \right \| ,\ \ \mathop{\rm det} | A _ {j} ^ {i} | \neq 0 , $$

and

$$ \left \| \begin{array}{cc} 0 &\mathfrak g ^ {i} \\ 0 &\mathfrak g _ {j} ^ {i} \\ \end{array} \right \| \ \ ( i , j = 1, \dots, n ) . $$

By the Cartan–Laptev theorem, the $ \mathfrak g $-valued $ 1 $-form

$$ \theta = \ \left \| \begin{array}{cc} 0 &\omega ^ {i} \\ 0 &\omega _ {j} ^ {i} \\ \end{array} \right \| $$

on $ P $ is the connection form of a certain affine connection on $ M $ if and only if

$$ d \omega ^ {i} + \omega _ {j} ^ {i} \wedge \omega ^ {j} = \ \frac{1}{2} T _ {jk} ^ { i } \omega ^ {j} \wedge \omega ^ {k} , $$

$$ d \omega _ {j} ^ {i} = \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} = \frac{1}{2} R _ {jkl} ^ {i} \omega ^ {k} \wedge \omega ^ {l} . $$

Here $ T _ {jk} ^ { i } $ and $ R _ {jkl} ^ {i} $ form, respectively, the torsion and curvature tensors of the affine connection on $ M $. The last two equations for the components of the connection form are called the structure equations for the affine connection on $ M $.

References

[1]	E. Cartan, "Espaces à connexion affine, projective et conforme" Acta Math. , 48 (1926) pp. 1–42
[2]	G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian)
[3]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)