# Connection form

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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 ; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev . 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 $M$.

How to Cite This Entry:
Connection form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection_form&oldid=46476
This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article