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In mathematics, a space, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature . In a space, the curvature is bounded from above by . A notable special case is ; complete spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard.

Originally, Aleksandrov called these spaces “ domain”. The terminology was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications).

DefinitionsEdit

 
Model triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.

For a real number  , let   denote the unique complete simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature  . Denote by   the diameter of  , which is   if   and   for  .

Let   be a geodesic metric space, i.e. a metric space for which every two points   can be joined by a geodesic segment, an arc length parametrized continuous curve  , whose length

 

is precisely  . Let   be a triangle in   with geodesic segments as its sides.   is said to satisfy the   inequality if there is a comparison triangle   in the model space  , with sides of the same length as the sides of  , such that distances between points on   are less than or equal to the distances between corresponding points on  .

The geodesic metric space   is said to be a   space if every geodesic triangle   in   with perimeter less than   satisfies the   inequality. A (not-necessarily-geodesic) metric space   is said to be a space with curvature   if every point of   has a geodesically convex   neighbourhood. A space with curvature   may be said to have non-positive curvature.

ExamplesEdit

  • Any   space   is also a   space for all  . In fact, the converse holds: if   is a   space for all  , then it is a   space.
  • The  -dimensional Euclidean space   with its usual metric is a   space. More generally, any real inner product space (not necessarily complete) is a   space; conversely, if a real normed vector space is a   space for some real  , then it is an inner product space.
  • The  -dimensional hyperbolic space   with its usual metric is a   space, and hence a   space as well.
  • The  -dimensional unit sphere   is a   space.
  • More generally, the standard space   is a   space. So, for example, regardless of dimension, the sphere of radius   (and constant curvature  ) is a   space. Note that the diameter of the sphere is   (as measured on the surface of the sphere) not   (as measured by going through the centre of the sphere).
  • The punctured plane   is not a   space since it is not geodesically convex (for example, the points   and   cannot be joined by a geodesic in   with arc length 2), but every point of   does have a   geodesically convex neighbourhood, so   is a space of curvature  .
  • The closed subspace   of   given by
 
equipped with the induced length metric is not a   space for any  .
  • Any product of   spaces is  . (This does not hold for negative arguments.)

Hadamard spacesEdit

As a special case, a complete CAT(0) space is also known as a Hadamard space; this is by analogy with the situation for Hadamard manifolds. A Hadamard space is contractible (it has the homotopy type of a single point) and, between any two points of a Hadamard space, there is a unique geodesic segment connecting them (in fact, both properties also hold for general, possibly incomplete, CAT(0) spaces). Most importantly, distance functions in Hadamard spaces are convex: if   are two geodesics in X defined on the same interval of time I, then the function   given by

 

is convex in t.

Properties of spacesEdit

Let   be a   space. Then the following properties hold:

  • Given any two points   (with   if  ), there is a unique geodesic segment that joins   to  ; moreover, this segment varies continuously as a function of its endpoints.
  • Every local geodesic in   with length at most   is a geodesic.
  • The  -balls in   of radius less than   are (geodesically) convex.
  • The  -balls in   of radius less than   are contractible.
  • Approximate midpoints are close to midpoints in the following sense: for every   and every   there exists a   such that, if   is the midpoint of a geodesic segment from   to   with   and
 
then  .
  • It follows from these properties that, for   the universal cover of every   space is contractible; in particular, the higher homotopy groups of such a space are trivial. As the example of the  -sphere   shows, there is, in general, no hope for a   space to be contractible if  .

See alsoEdit

ReferencesEdit

  • Alexander, Stephanie; Kapovitch, Vitali; Petrunin, Anton. "Alexandrov Geometry, Chapter 7" (PDF). Retrieved 2011-04-07.
  • Alexander, Stephanie; Kapovitch, Vitali; Petrunin, Anton. "Invitation to Alexandrov geometry: CAT[0] spaces". arXiv:1701.03483 [math.DG].
  • Ballmann, Werner (1995). Lectures on spaces of nonpositive curvature. DMV Seminar 25. Basel: Birkhäuser Verlag. pp. viii+112. ISBN 3-7643-5242-6. MR 1377265.
  • Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319. Berlin: Springer-Verlag. pp. xxii+643. ISBN 3-540-64324-9. MR 1744486.
  • Gromov, Mikhail (1987). "Hyperbolic groups". Essays in group theory. Math. Sci. Res. Inst. Publ. 8. New York: Springer. pp. 75–263. MR 0919829.
  • Hindawi, Mohamad A. (2005). Asymptotic invariants of Hadamard manifolds (PDF). University of Pennsylvania: PhD thesis.