Ehresmann connection
E542125
An Ehresmann connection is a geometric structure on a fiber bundle that specifies a way to consistently split tangent spaces into vertical and horizontal parts, enabling the definition of parallel transport.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
connection on a fiber bundle
ⓘ
geometric structure ⓘ |
| appearsIn |
geometric theory of ordinary differential equations
ⓘ
modern formulations of gauge fields ⓘ submersion theory ⓘ |
| appliesTo |
general smooth fiber bundles
ⓘ
principal bundles ⓘ vector bundles ⓘ |
| characterizedBy |
choice of horizontal subspace at each point of the total space
ⓘ
smooth variation of horizontal subspaces ⓘ |
| contrastsWith |
Levi-Civita connection
NERFINISHED
ⓘ
linear connection defined via covariant derivative on sections ⓘ |
| curvatureMeasures | non-integrability of the horizontal distribution ⓘ |
| definedOn | fiber bundle ⓘ |
| defines |
horizontal lift of curves
ⓘ
horizontal lift of vector fields ⓘ |
| enables | parallel transport along curves in the base space ⓘ |
| formalizedAs | smooth horizontal distribution complementary to the vertical distribution ⓘ |
| generalizes |
affine connection on a manifold
ⓘ
linear connection on a vector bundle ⓘ |
| hasComponent |
horizontal distribution
ⓘ
vertical distribution ⓘ |
| hasCurvature | Ehresmann curvature NERFINISHED ⓘ |
| hasProperty |
horizontal subspaces are complementary to vertical subspaces
ⓘ
vertical subspaces are kernels of the differential of the bundle projection ⓘ |
| integrableIf | horizontal distribution is tangent to a foliation by local sections ⓘ |
| is | connection concept independent of linear structure on fibers ⓘ |
| namedAfter | Charles Ehresmann NERFINISHED ⓘ |
| relatedTo |
connection 1-form
ⓘ
curvature of a connection ⓘ principal connection ⓘ |
| requires |
smooth bundle projection
ⓘ
smooth structure on base space ⓘ smooth structure on total space ⓘ |
| splits | tangent bundle of the total space of a fiber bundle ⓘ |
| splitsInto |
horizontal subbundle
ⓘ
vertical subbundle ⓘ |
| studiedIn | Ehresmann’s theory of connections and fiber spaces ⓘ |
| usedFor |
defining geodesic-like curves in fiber bundles
ⓘ
lifting symmetries from base to total space ⓘ |
| usedIn |
category theory inspired geometry
ⓘ
differential geometry ⓘ gauge theory ⓘ global analysis ⓘ theory of foliations ⓘ |
| usedToDefine |
covariant derivative along curves
ⓘ
holonomy of a fiber bundle ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Cartan connection