Kobayashi metric
E521038
The Kobayashi metric is an intrinsic pseudometric in complex analysis that measures hyperbolic distance on complex manifolds and generalizes the Poincaré metric to higher dimensions.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
hyperbolic metric
ⓘ
intrinsic metric ⓘ invariant metric in complex analysis ⓘ pseudometric ⓘ |
| agreesWith | Hilbert metric on the unit ball in C^n up to equivalence ⓘ |
| appearsIn | "Hyperbolic Manifolds and Holomorphic Mappings" by Shoshichi Kobayashi NERFINISHED ⓘ |
| appliesTo |
complex analytic spaces
ⓘ
complex manifolds ⓘ |
| coincidesWith |
Carathéodory metric on the unit disc
ⓘ
Poincaré metric on the unit disc ⓘ |
| definedUsing |
chains of holomorphic discs
ⓘ
infimum of lengths of chains of analytic discs ⓘ |
| field |
complex analysis
ⓘ
complex differential geometry ⓘ several complex variables ⓘ |
| generalizes | Poincaré metric NERFINISHED ⓘ |
| introducedBy | Shoshichi Kobayashi NERFINISHED ⓘ |
| introducedIn | 1960s ⓘ |
| isCompleteOn |
bounded convex domains in C^n
ⓘ
bounded strongly pseudoconvex domains ⓘ |
| isContractedBy | holomorphic self-maps ⓘ |
| isDistanceDecreasingUnderHolomorphicMaps | true ⓘ |
| isEquivalentTo |
Bergman metric on certain bounded symmetric domains
ⓘ
Carathéodory metric on taut domains ⓘ |
| isFinslerType | true ⓘ |
| isHolomorphicallyInvariant | true ⓘ |
| isIntrinsic | true ⓘ |
| isInvariantUnder | biholomorphic maps ⓘ |
| isLargestPseudometricWithDistanceDecreasingProperty | true ⓘ |
| isLocalFinslerMetricOn | tangent bundle of a complex manifold ⓘ |
| isMonotoneUnder | holomorphic mappings ⓘ |
| isNondegenerateOn | Kobayashi hyperbolic manifolds ⓘ |
| isPseudometric | true ⓘ |
| isToolFor |
proving Brody hyperbolicity
ⓘ
studying negative curvature in complex geometry ⓘ |
| isUpperSemicontinuous | true ⓘ |
| mayFailToSeparatePoints | true ⓘ |
| namedAfter | Shoshichi Kobayashi NERFINISHED ⓘ |
| relatedConcept |
Bergman metric
ⓘ
Carathéodory metric NERFINISHED ⓘ Kobayashi hyperbolicity NERFINISHED ⓘ Teichmüller metric NERFINISHED ⓘ |
| satisfies | Schwarz–Pick type inequalities ⓘ |
| usedToStudy |
complex dynamical systems
ⓘ
holomorphic mappings between complex manifolds ⓘ hyperbolicity properties of complex manifolds ⓘ |
| vanishesIdenticallyOn |
complex Euclidean space C^n
ⓘ
complex projective space ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.