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A theorem of Archimedes about spheres and cylinders and two-point homogeneous spaces

Published online by Cambridge University Press:  17 April 2009

M. Djorić  and
L. Vanhecke
M. Djorić
Affiliation:
Faculty of Mathematics, University of Belgrade, P.B. 550 Studentski trg 16 11000 Belgrade, Yugoslavia
L. Vanhecke
Affiliation:
Department of Mathematics Katholieke, Universiteit Leuven Celestijnenlaan, 200B B-3001 Leuven, Belgium
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Abstract

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Starting from the well-known classical theorem of Archimedes about the volumes of spheres and circumscribing cylinders in three-dimensional Euclidean space, one considers circumscribing tubes of small geodesic spheres in general Riemannian manifolds and one derives new characterisations of two-point homogeneous spaces from it.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 2 , April 1991 , pp. 283 - 294
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Chen, B.Y. and Vanhecke, L., ‘Differential geometry of geodesic spheres’, J. Reine Angew. Math. 325 (1981), 28–67.Google Scholar
[2]Gray, A., ‘The volume of a small geodesic ball in a Riemannian manifold’, Michigan Math. J. 20 (1973), 329–344.Google Scholar
[3]Gray, A. and Vanhecke, L., ‘Riemannian geometry as determined by the volumes of small geodesic balls’, Acta Math. 142 (1979), 157–198.CrossRefGoogle Scholar
[4]Gray, A. and Vanhecke, L., ‘Oppervlakten van geodetische cirkels op oppervlakken’, Med. Konink. Acad. Wetensch. België 42 (1980), 1–17.Google Scholar
[5]Gray, A. and Vanhecke, L., ‘The volumes of tubes in a Riemannian manifold’, Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), 1–50.Google Scholar
[6]Gray, A. and Vanhecke, L., ‘The volumes of tubes about curves in a Riemannian manifold’, Proc. London Math. Soc. 44 (1982), 215–243.CrossRefGoogle Scholar
[7]Gray, A., Tubes (Addison-Wesley Publ. Co., Reading, 1989).Google Scholar
[8]Kowalski, O. and Vanhecke, L., ‘Ball-homogeneous and disk-homogeneous Riemannian manifolds’, Math. Z. 180 (1982), 429–444.CrossRefGoogle Scholar
[9]Kowalski, O. and Vanhecke, L., ‘On disk-homogeneous symmetric spaces’, Ann. Global Anal. Geom. 1 (1983), 91–104.CrossRefGoogle Scholar
[10]Kowalski, O. and Vanhecke, L., ‘G-deformations of curves and volumes of tubes in Riemannian manifolds’, Geom. Dedicata 15 (1983), 125–135.CrossRefGoogle Scholar
[11]Kowalski, O. and Vanhecke, L., ‘The volume of geodesic disks in a Riemannian manifold’, Czechoslovak Math. J. 35 (1985), 66–77.CrossRefGoogle Scholar
[12]Lutwak, E., ‘On isoperimetric inequalities related to a problem of Moser’, Amer. Math. Monthly 86 (1979), 476–477.CrossRefGoogle Scholar
[13]Gomis, S. Segura, Teoremas de comparación ligados al problema isoperimétrico, Tesis Doctoral (Departamento de Geometria y Topologia, Universidad de Valencia, Valencia, 1987).Google Scholar
[14]Vanhecke, L., Geometry in Normal and Tubular Neighborhoods, Lecture Notes (Department of Mathematics, University of Leuven, Leuven, 1988). Proc. Workshop on Differential Geometry and Topology (Cala Gonone, Sardinia 1988). (to appear).Google Scholar