Pretangent spaces with nonpositive and nonnegative Aleksandrov curvature

V. V. Bilet; O. A. Dovgoshey

We find conditions under which the pretangent spaces to general metric spaces have the nonpositive Aleksandrov curvature or nonnegative one. The infinitesimal structure of general metric spaces with Busemann convex pretangent spaces is also described.

1. F. Abdullayev, O. Dovgoshey, M. Kucukaslan, Compactness and boundedness of tangent spaces to metric spaces, Beitr. Algebra Geom., 51 (2010), №2, 547–576.

2. I. Berg, I. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218.

3. V. Bilet, Geodesic tangent spaces to metric spaces, Ukr. Mat. J., 64 (2012), №9, 1273–1281. (in Russian)

4. V. Bilet, O. Dovgoshey, Isometric embeddings of pretangent spaces in En, Bull. Belg. Math. Soc. Simon Stevin., 20 (2013), 91–110.

5. V. Bilet, O. Dovgoshey, Metric betweenness, Ptolemaic spaces and isometric embeddings of pretangent spaces in R, J. Math. Sci., New York, 182 (2012), №4, 22–36.

6. D. Burago, Yu. Burago, S. Ivanov, A course in metric geometry, American Mathematical Society, Providence, RI, 2001.

7. D. Dordovskyi, Metric tangent spaces to Euclidean spaces, J. Math. Sci., New York, 179 (2011), №2, 229–244.

8. O. Dovgoshey, O. Martio, Tangent spaces to metric spaces, Reports in Math., Helsinki Univ., 480 (2008), 20 p.

9. O. Dovgoshey, O. Martio, Tangent spaces to general metric spaces, Rev. Roumaine Math. Pures. Appl., 56 (2011), №2, 137–155.

10. T. Foertsch, A. Lytchak, V. Schroeder, Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not., (2007), №22, 15 p.

11. N. Lebedeva, A. Petrunin, Curvature bounded below: a definition a la Berg-Nikolaev, Electronic research announcements in math. sci., 17 (2010), 122–124.

12. A. Papadopoulos, Metric spaces, convexity and nonpositive curvature, European Mathematical Society, (2005).

13. T. Sato, An alternative proof of Berg and Nikilaev’s characterization of CAT(0)-spaces via quadrialateral inequality, Arch. Math., 93 (2009), 487–490.

	Pretangent spaces with nonpositive and nonnegative Aleksandrov curvature (in Russian)
Author	V. V. Bilet, O. A. Dovgoshey biletvictoriya@mail.ru, aleksdov@mail.ru Donetsk Institute of Applied Mathematics and Mechanics
Abstract	We find conditions under which the pretangent spaces to general metric spaces have the nonpositive Aleksandrov curvature or nonnegative one. The infinitesimal structure of general metric spaces with Busemann convex pretangent spaces is also described.
Keywords	pretangent space; CAT(0)-space; Aleksandrov curvature; Busemann convexity; infinitesimal geometry of metric spaces
Reference	1. F. Abdullayev, O. Dovgoshey, M. Kucukaslan, Compactness and boundedness of tangent spaces to metric spaces, Beitr. Algebra Geom., 51 (2010), №2, 547–576. 2. I. Berg, I. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218. 3. V. Bilet, Geodesic tangent spaces to metric spaces, Ukr. Mat. J., 64 (2012), №9, 1273–1281. (in Russian) 4. V. Bilet, O. Dovgoshey, Isometric embeddings of pretangent spaces in En, Bull. Belg. Math. Soc. Simon Stevin., 20 (2013), 91–110. 5. V. Bilet, O. Dovgoshey, Metric betweenness, Ptolemaic spaces and isometric embeddings of pretangent spaces in R, J. Math. Sci., New York, 182 (2012), №4, 22–36. 6. D. Burago, Yu. Burago, S. Ivanov, A course in metric geometry, American Mathematical Society, Providence, RI, 2001. 7. D. Dordovskyi, Metric tangent spaces to Euclidean spaces, J. Math. Sci., New York, 179 (2011), №2, 229–244. 8. O. Dovgoshey, O. Martio, Tangent spaces to metric spaces, Reports in Math., Helsinki Univ., 480 (2008), 20 p. 9. O. Dovgoshey, O. Martio, Tangent spaces to general metric spaces, Rev. Roumaine Math. Pures. Appl., 56 (2011), №2, 137–155. 10. T. Foertsch, A. Lytchak, V. Schroeder, Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not., (2007), №22, 15 p. 11. N. Lebedeva, A. Petrunin, Curvature bounded below: a definition a la Berg-Nikolaev, Electronic research announcements in math. sci., 17 (2010), 122–124. 12. A. Papadopoulos, Metric spaces, convexity and nonpositive curvature, European Mathematical Society, (2005). 13. T. Sato, An alternative proof of Berg and Nikilaev’s characterization of CAT(0)-spaces via quadrialateral inequality, Arch. Math., 93 (2009), 487–490.
Pages	198-208
Volume	40
Issue	2
Year	2013
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Pretangent spaces with nonpositive and nonnegative Aleksandrov curvature (in Russian)