Alexandrov space
In today's world, Alexandrov space is a topic that has captured the attention of millions of people around the world. Whether due to its impact on society, its historical relevance or its influence on people's daily lives, Alexandrov space has managed to become a recurring topic in conversations, debates and discussions in all areas. With a constant presence in the media and social networks, Alexandrov space has managed to transcend borders and cultures, generating interest and concern in people of all ages and conditions. In this article, we will thoroughly explore the impact and importance of Alexandrov space, analyzing its implications and relevance in contemporary society.
In geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of 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" (see Comparison triangle#Alexandrov angles) 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 Aleksandr Danilovich Aleksandrov in 1948[3] and should not be confused with Alexandrov-discrete spaces named after the Russian topologist Pavel Alexandrov. They were studied in detail by Burago, Gromov and Perelman in 1992[4] and were later used in Perelman's proof of the Poincaré conjecture.
References
- ^ 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.
- ^ Aleksandrov, A D; Berestovskii, V N; Nikolaev, I G (1986-01-01). "Generalized Riemannian spaces". Russian Mathematical Surveys. 41 (3): 1–54. doi:10.1070/rm1986v041n03abeh003311. ISSN 0036-0279.
- ^ a b Berger, Marcel (2003). A Panoramic View of Riemannian Geometry. Springer. p. 704.
- ^ Burago, Yuri; Gromov, Mikhail Leonidovich; Perelman, Grigori (1992). "A.D. Alexandrov spaces with curvature bounded below". Russian Math. Surveys. 47 (2): 1–58. doi:10.1070/RM1992v047n02ABEH000877.
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