Difference between revisions of "Connection object"
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− | + | A differential-geometric object on a smooth principal fibre bundle | |
+ | that is used to define a [[Horizontal distribution|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|connection form]] \theta , | |
+ | given by the relation $ \theta ^ \rho = \omega ^ \rho - \Gamma _ {i} ^ \rho \omega ^ {i} $, | ||
+ | and its [[Curvature form|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 | 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(2)|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. | Here this follows from the condition of invariance under displacement. |
Latest revision as of 17:46, 4 June 2020
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.
Connection object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection_object&oldid=13997