|
Curvature, triameter, and beyond
Author(s):
Karsten
Grove;
Steen
Markvorsen
Journal:
Bull. Amer. Math. Soc.
27
(1992),
261-265.
MSC (2000):
Primary 53C23;
Secondary 53C20
MathSciNet review:
1152160
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
In its most general form, the recognition problem in Riemannian geometry asks for the identification of an unknown Riemannian manifold via measurements of metric invariants on the manifold. We introduce a new infinite sequence of invariants, the first term of which is the usual diameter, and illustrate the role of these global shape invariants in a number of recognition problems.
References:
-
- [BGP]
- Y. Burago, M. Gromov, and G. Perelman, Aleksandrov's spaces with curvatures bounded from below I, Uspekhi Mat. Nauk (to appear).
- [CE]
- J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Math Library, vol. 9, North-Holland, Amsterdam, 1975. MR 0458335 (56:16538)
- [GG]
- D. Gromoll and K. Grove, A generalization of Berger 's rigidity theorem for positively curved manifolds, Ann. Sci. École. Norm. Sup. 20 (1987), 227-239. MR 911756 (88k:53062)
- [G]
- M. Gromov, Filling Riemannian Manifolds, J. Differential Geom. 18 (1983), 1-148. MR 697984 (85h:53029)
- [G1]
- K. Grove, Metric differential geometry, Differential Geometry (V. L. Hansen, ed.), Lecture Notes in Math, vol. vol. 1263, Springer-Verlag, 1987, pp. 171-227. MR 905882 (88i:53075)
- [G2]
- -, Metric and topological measurements of manifolds, Proc. International Congress of Mathematicians, Kyoto, Japan 1990, Springer-Verlag, 1991, pp. 511-519. MR 1159238 (93c:53030)
- [GM]
- K. Grove and S. Markvorsen, Metric invariants for the Riemannian recognition program via Aleksandrov geometry, preprint.
- [GP1]
- K. Grove and P. Petersen V, Homotopy types of positively curved manifolds with large volume, Amer. J. Math. 110 (1988), 1183-1188. MR 970124 (89m:53063)
- [GP2]
- -, A pinching theorem for homotopy spheres, J. Amer. Math. Soc. 3 (1990), 671-677. MR 1049696 (91e:53040)
- [GP3]
- -, Manifolds near the boundary of existence, J. Differential Geom. 33 (1991), 379-394. MR 1094462 (92a:53067)
- [GP4]
- -, Volume comparison à la Aleksandrov, Acta. Math. (to appear).
- [GPW]
- K. Grove, P. Petersen, and J.-Y. Wu, Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990), 205-213. MR 1029396 (90k:53075)
- [GS]
- K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), 201-211. MR 0500705 (58:18268)
- [H]
- E. Hille, Remarks on transfinite diameters, J. Analyse Math. 14 (1965), 209-224. MR 0199690 (33:7833)
- [K1]
- M. Katz, The filling radius of two-point homogeneous spaces, J. Differential Geom. 18 (1983), 505-511. MR 723814 (85h:53030)
- [K2]
- -, The rational filling radius of complex projective spaces, preprint.
- [K3]
- -, The first diameters of 3-manifolds of positive scalar curvature, Proc. Amer. Math. Soc. 104 (1988), 591-595. MR 962834 (89j:53040)
- [N]
- F. Nielsen, On the sum of distances between n points on the sphere (Danish), Nordisk Tidskr. Mat. 13 (1965), 45-50.
- [O]
- Y. Otsu, On manifolds of small excess, Amer. J. Math. (to appear). MR 1254733 (95i:53046)
- [OSY]
- Y. Otsu, K. Shiohama, and T. Yamaguchi, A new version of differentiable sphere theorem, Invent. Math. 98 (1989), 219-228. MR 1016261 (90i:53049)
- [PZ]
- P. Petersen and Zhu, An excess sphere theorem, Ann. Sci. Éc. Norm. Sup. (to appear).
- [R]
- W. Rinow, Die Innere Geometrie der Metrischen Räume (1961), Springer-Verlag, 1961. MR 0123969 (23:A1290)
- [SY]
- K. Shiohama and T. Yamaguchi, Positively curved manifolds with restricted diameters, Geometry of Manifolds (K. Shiohama, ed.), Perspect. in Math, vol. 8, Academic Press, Boston, MA, 1989, pp. 345-350. MR 1040534 (90m:53056)
- [T]
- F. Toth, On the sum of distances determined by a pointset, Acta. Math. Acad. Sci. Hungar. 7 (1957), 397-401. MR 0107212 (21:5937)
- [U]
- P. Urysohn, Notes supplémentaires, Fund. Math. 8 (1926), 352-356.
- [W]
- F. Wilhelm, On the filling radius of positively curved manifolds, Invent. Math. 107 (1992), 653-668. MR 1150606 (93d:53055)
- [Y]
- T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991), 317-357. MR 1097241 (92b:53067)
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(2000):
53C23, 53C20
Retrieve articles in all Journals with MSC
(2000):
53C23, 53C20
Additional Information:
DOI:
10.1090/S0273-0979-1992-00308-7
PII:
S 0273-0979(1992)00308-7
Copyright of article:
Copyright
1992,
American Mathematical Society
|
