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Geodesic bicombing

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In metric geometry, a geodesic bicombing distinguishes a class of geodesics of a metric space. The study of metric spaces with distinguished geodesics traces back to the work of the mathematician Herbert Busemann. The convention to call a collection of paths of a metric space bicombing is due to William Thurston. By imposing a weak global non-positive curvature condition on a geodesic bicombing several results from the theory of CAT(0) spaces and Banach space theory may be recovered in a more general setting.

Definition

Let ( X , d ) {\displaystyle (X,d)} be a metric space. A map σ : X × X × [ 0 , 1 ] → X {\displaystyle \sigma \colon X\times X\times \to X} is a geodesic bicombing if for all points x , y ∈ X {\displaystyle x,y\in X} the map σ x y ( ⋅ ) := σ ( x , y , ⋅ ) {\displaystyle \sigma _{xy}(\cdot ):=\sigma (x,y,\cdot )} is a unit speed metric geodesic from x {\displaystyle x} to y {\displaystyle y} , that is, σ x y ( 0 ) = x {\displaystyle \sigma _{xy}(0)=x} , σ x y ( 1 ) = y {\displaystyle \sigma _{xy}(1)=y} and d ( σ x y ( s ) , σ x y ( t ) ) = | s − t | d ( x , y ) {\displaystyle d(\sigma _{xy}(s),\sigma _{xy}(t))=\vert s-t\vert d(x,y)} for all real numbers s , t ∈ [ 0 , 1 ] {\displaystyle s,t\in } .

Different classes of geodesic bicombings

A geodesic bicombing σ : X × X × [ 0 , 1 ] → X {\displaystyle \sigma \colon X\times X\times \to X} is:

  • reversible if σ x y ( t ) = σ y x ( 1 − t ) {\displaystyle \sigma _{xy}(t)=\sigma _{yx}(1-t)} for all x , y ∈ X {\displaystyle x,y\in X} and t ∈ [ 0 , 1 ] {\displaystyle t\in } .
  • consistent if σ x y ( ( 1 − λ ) s + λ t ) = σ p q ( λ ) {\displaystyle \sigma _{xy}((1-\lambda )s+\lambda t)=\sigma _{pq}(\lambda )} whenever x , y ∈ X , 0 ≤ s ≤ t ≤ 1 , p := σ x y ( s ) , q := σ x y ( t ) , {\displaystyle x,y\in X,0\leq s\leq t\leq 1,p:=\sigma _{xy}(s),q:=\sigma _{xy}(t),} and λ ∈ [ 0 , 1 ] {\displaystyle \lambda \in } .
  • conical if d ( σ x y ( t ) , σ x ′ y ′ ( t ) ) ≤ ( 1 − t ) d ( x , x ′ ) + t d ( y , y ′ ) {\displaystyle d(\sigma _{xy}(t),\sigma _{x^{\prime }y^{\prime }}(t))\leq (1-t)d(x,x^{\prime })+td(y,y^{\prime })} for all x , x ′ , y , y ′ ∈ X {\displaystyle x,x^{\prime },y,y^{\prime }\in X} and t ∈ [ 0 , 1 ] {\displaystyle t\in } .
  • convex if t ↦ d ( σ x y ( t ) , σ x ′ y ′ ( t ) ) {\displaystyle t\mapsto d(\sigma _{xy}(t),\sigma _{x^{\prime }y^{\prime }}(t))} is a convex function on [ 0 , 1 ] {\displaystyle } for all x , x ′ , y , y ′ ∈ X {\displaystyle x,x^{\prime },y,y^{\prime }\in X} .

Examples

Examples of metric spaces with a conical geodesic bicombing include:

Properties

  • Every consistent conical geodesic bicombing is convex.
  • Every convex geodesic bicombing is conical, but the reverse implication does not hold in general.
  • Every proper metric space with a conical geodesic bicombing admits a convex geodesic bicombing.
  • Every complete metric space with a conical geodesic bicombing admits a reversible conical geodesic bicombing.

References

  1. Busemann, Herbert (1905-) (1987). Spaces with distinguished geodesics. Dekker. ISBN 0-8247-7545-7. OCLC 908865701.{{cite book}}: CS1 maint: numeric names: authors list (link)
  2. Epstein, D. B. A. (1992). Word processing in groups. Jones and Bartlett Publishers. p. 84. ISBN 0-86720-244-0. OCLC 911329802.
  3. ^ Descombes, Dominic; Lang, Urs (2015). "Convex geodesic bicombings and hyperbolicity". Geometriae Dedicata. 177 (1): 367–384. doi:10.1007/s10711-014-9994-y. hdl:20.500.11850/87627. ISSN 0046-5755.
  4. Basso, Giuliano; Miesch, Benjamin (2019). "Conical geodesic bicombings on subsets of normed vector spaces". Advances in Geometry. 19 (2): 151–164. arXiv:1604.04163. doi:10.1515/advgeom-2018-0008. hdl:20.500.11850/340286. ISSN 1615-7168. S2CID 15595365.
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