Alexandrov space

Source: Wikipedia, the free encyclopedia.

In

geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces.[1][2]

One can show that the Hausdorff dimension of an Alexandrov space with curvature ≥ k is either a non-negative integer or infinite.[1] One can define a notion of "angle" and "tangent cone" in these spaces.

Alexandrov spaces with curvature ≥ k are important as they form the limits (in the

Gromov-Hausdorff metric) of sequences of Riemannian manifolds with sectional curvature ≥ k,[3] as described by Gromov's compactness theorem
.

Alexandrov spaces with curvature ≥ k were introduced by the Russian mathematician

Gromov and Perelman in 1992[4] and were later used in Perelman's proof of the Poincaré conjecture
.

References

  1. ^ a b Kathusiro Shiohama (July 13–17, 1992). An Introduction to the Geometry of Alexandrov Spaces (PDF). Daewoo Workshop on Differential Geometry. Kwang Won University, Chunchon, Korea.
  2. ISSN 0036-0279
    .
  3. ^ a b Berger, Marcel (2003). A Panoramic View of Riemannian Geometry. Springer. p. 704.
  4. doi:10.1070/RM1992v047n02ABEH000877
    .
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