Busemann function
E855796
The Busemann function is a geometric tool in metric and Riemannian geometry that measures asymptotic distance along geodesic rays, often used to study the large-scale structure and boundaries of spaces.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
function in metric geometry
ⓘ
mathematical concept ⓘ tool in Riemannian geometry ⓘ |
| appearsIn | Busemann’s work on the geometry of geodesics ⓘ |
| associatedWith |
asymptotic class of geodesic rays
ⓘ
geodesic ray ⓘ horofunction ⓘ ideal boundary point ⓘ |
| category | asymptotic invariants of metric spaces ⓘ |
| codomain | real numbers ⓘ |
| definition | given a geodesic ray c(t) and basepoint x, b_c(x)=lim_{t→∞}(d(x,c(t))−t) when the limit exists NERFINISHED ⓘ |
| dependsOn | choice of geodesic ray ⓘ |
| domain |
Riemannian manifold
ⓘ
geodesic metric space ⓘ |
| field |
Riemannian geometry
NERFINISHED
ⓘ
geometric group theory ⓘ global differential geometry ⓘ metric geometry ⓘ |
| generalizes | linear functionals in normed spaces ⓘ |
| invariantUnder | reparametrization of the geodesic ray by additive constants ⓘ |
| namedAfter | Herbert Busemann NERFINISHED ⓘ |
| property |
1-Lipschitz in CAT(0) spaces
ⓘ
affine along geodesics in normed vector spaces ⓘ convex along geodesics in CAT(0) spaces ⓘ harmonic in simply connected complete manifolds of constant negative curvature ⓘ |
| relatedTo |
Gromov boundary
NERFINISHED
ⓘ
horofunction boundary NERFINISHED ⓘ visual boundary ⓘ |
| satisfies | triangle inequality type estimates derived from the metric ⓘ |
| usedFor |
defining Busemann boundary
ⓘ
defining Busemann compactification ⓘ defining geometric boundaries of spaces ⓘ describing horoballs ⓘ describing horospheres ⓘ measuring asymptotic distance along geodesic rays ⓘ studying Gromov hyperbolic spaces ⓘ studying large-scale structure of metric spaces ⓘ studying nonpositive curvature ⓘ studying visibility properties of spaces ⓘ |
| usedIn |
asymptotic geometry of manifolds
ⓘ
construction of Patterson–Sullivan measures ⓘ ergodic theory on negatively curved manifolds ⓘ potential theory on manifolds ⓘ study of CAT(0) spaces ⓘ study of Gromov hyperbolic groups ⓘ study of Hadamard manifolds ⓘ study of geodesic flow ⓘ |
| wellDefinedUpTo | additive constant ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.