# CURVATURE BOUNDS OF EUCLIDEAN CONES OF SPHERES

• Chai, Y.D. (Department of Mathematics, Sungkyunkwan University) ;
• Kim, Yong-Il (School of Media, Soongsil University) ;
• Lee, Doo-Hann (Department of Mathematics, Sungkyunkwan University)
• Published : 2003.05.01
• 49 5

#### Abstract

In this paper, we obtain the optimal condition of the curvature bounds guaranteeing that Euclidean cones over Aleksandrov spaces of curvature bounded above preserve the curvature bounds, by considering the Euclidean cone CS$_{r}$ $^{n}$ over n-dimensional sphere S$_{r}$ $^{n}$ of radius r. More precisely, we show that for r<1, the Euclidean cone CS$_{r}$ $^{n}$ of S$_{r}$ $^{n}$ is a CBB(0) space, but not a CBA($textsc{k}$)-space for any real $textsc{k}$$\in$R.

#### Keywords

Euclidean cone;Aleksandrov spaces;interior metrics

#### References

1. Lecture notes on spring semester 1994/95 academic year Lectures on spaces of curvature bounded above S.Buyalo
2. Russian Math. Surveys v.41 no.3 Generalized Riemannian spaces A.D.Aleksandrov;V.N.Berestovskii;I.G.Nikolaev
3. Russian Math. Surveys v.47 no.2 A. D. Alexandrov's spaces with curvature bounded below Y.Burago;M.Gromov;G.Perel'man https://doi.org/10.1070/RM1992v047n02ABEH000877
4. Manuscripta math. v.86 The tangent cone of an Alexandrov space of curvature ≤ K I.Nikolaev https://doi.org/10.1007/BF02567983
5. Ann. Global Anal. Geom. v.15 Jung's theorem for Alexandrov spaces of curvature bounded above U.Lang;V.Schroeder https://doi.org/10.1023/A:1006574402955

#### Cited by

1. The Deformation Spaces of Projective Structures on 3-Dimensional Coxeter Orbifolds vol.119, pp.1, 2006, https://doi.org/10.1007/s10711-006-9050-7
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