Connection object

A differential-geometric object on a smooth principal fibre bundle $ P $ that is used to define a horizontal distribution $ \Delta $ of a connection in $ P $. Let $ R _ {0} ( P) $ be the bundle of all tangent frames to $ P $ such that the first $ r $ vectors $ e _ {1} \dots e _ {r} $ are tangent to the corresponding fibre, and are generated by $ r $ basis elements in the Lie algebra of the structure group $ G $ of $ P $, $ r = \mathop{\rm dim} G $. A connection object then consists of functions $ \Gamma _ {i} ^ \rho $ on $ R _ {0} ( P) $ such that the subspace of $ \Delta $ is spanned by the vectors $ e _ {i} + \Gamma _ {i} ^ \rho e _ \rho $ $ ( \rho , \sigma = 1 \dots r; i , j , \dots = r + 1 \dots r+ n ) $. Furthermore, the $ \Gamma _ {i} ^ \rho $ must satisfy the following conditions on $ R _ {0} ( P) $:

$$ \tag{1 } d \Gamma _ {i} ^ \rho - \Gamma _ {j} ^ \rho \omega _ {i} ^ {j} + \Gamma _ {i} ^ \sigma \omega _ \sigma ^ \rho + \omega _ {i} ^ \rho = \Gamma _ {ij} ^ \rho \omega ^ {j} . $$

They are expressed by using the $ 1 $- forms on $ R _ {0} ( P) $ that occur in the structure equations for the forms $ \omega ^ {i} , \omega ^ \rho $ given by the co-basis dual to $ \{ e _ {i} , e _ \rho \} $;

$$ \tag{2 } \left . \begin{array}{c} d \omega ^ {i} = \omega ^ {j} \wedge \omega _ {j} ^ {i} , \\ d \omega ^ \rho = - \frac{1}{2} C _ {\sigma \tau } ^ \rho \omega ^ \sigma \wedge \omega ^ \tau + \omega ^ {i} \wedge \omega _ {i} ^ \rho , \\ \omega _ \sigma ^ \rho = - C _ {\sigma \tau } ^ \rho \omega ^ \tau . \\ \end{array} \right \} $$

A connection object also defines a corresponding connection form $ \theta $, given by the relation $ \theta ^ \rho = \omega ^ \rho - \Gamma _ {i} ^ \rho \omega ^ {i} $, and its curvature form $ \Omega $, given by the formulas:

$$ \Omega ^ \rho = - \frac{1}{2} R _ {ij} ^ \rho \omega ^ {i} \wedge \omega ^ {j} , $$

$$ R _ {ij} ^ \rho = - 2 ( \Gamma _ {[ ij ] } ^ \rho + C _ {\sigma \tau } ^ \rho \Gamma _ {i} ^ \sigma \Gamma _ {j} ^ \tau ) . $$

For example, let $ P $ be the space of affine tangent frames of an $ n $- dimensional smooth manifold $ M $. Then the second equation in (2) has the form

$$ d \omega _ {j} ^ {i} = \ - \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} + \omega ^ {k} \wedge \omega _ {jk} ^ {i} $$

and (1) reduces to

$$ d \Gamma _ {ik} ^ {j} - \Gamma _ {lk} ^ {j} \omega _ {i} ^ {l} - \Gamma _ {il} ^ {j} \omega _ {k} ^ {l} + \Gamma _ {ik} ^ {l} \omega _ {l} ^ {j} + \omega _ {ik} ^ {j} = \ \Gamma _ {jkl} ^ {i} \omega ^ {l} . $$

Under parallel displacement one must have $ \omega _ {j} ^ {i} - \Gamma _ {jk} ^ {i} \omega ^ {k} = 0 $. If a local chart is chosen in $ M $, and if in its domain one makes the transition to the natural frame of the chart, i.e. $ \omega ^ {k} = dx ^ {k} $, then the parallel displacement is defined by $ \omega _ {j} ^ {i} = \Gamma _ {jk} ^ {i} dx ^ {k} $. The classical definition of a connection object of an affine connection on $ M $ is given by the set of functions $ \Gamma _ {jk} ^ {i} $ defined on the domains of the charts such that under transition to the coordinates of another chart these functions are transformed according to the formulas

$$ \Gamma _ {st} ^ { \prime r } = \ \frac{\partial x ^ {\prime r } }{\partial x ^ {i} } \frac{\partial x ^ {j} }{\partial x ^ {\prime s } } \frac{\partial x ^ {k} }{\partial x ^ {\prime t } } \Gamma _ {jk} ^ {i} + \frac{\partial ^ {2} x ^ {i} }{\partial x ^ {\prime s } \partial x ^ {\prime t } } \frac{\partial x ^ {\prime r } }{\partial x ^ {i} } . $$

Here this follows from the condition of invariance under displacement.