# 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.

How to Cite This Entry:
Connection object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection_object&oldid=46477
This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article