Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-28T16:19:12.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Divergence-preserving geodesic transformations

Published online by Cambridge University Press:  14 November 2011

E. García-Río  and
L. Vanhecke
E. García-Río
Affiliation:
Facultade de Matemáticas, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain e-mail: eduardo@zmat.usc.es
L. Vanhecke
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium e-mail: lieven.vanhecke@wis.kuleuven.ac.be
Get access Rights & Permissions [Opens in a new window]

Extract

We discuss divergence- and volume-preserving geodesic transformations with respect to submanifolds and in particular, with respect to hypersurfaces. We use these transformations to derive characterisations of special classes of hypersurfaces such as isoparametric hypersurfaces and Hopf hypersurfaces with constant principal curvatures. Furthermore, we consider divergence-preserving geodesic transformations with respect to geodesic spheres.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 6 , 1998 , pp. 1309 - 1323
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Berndt, J.. Über Untermannigfaltigkeiten von komplexen Raumformen (Doctoral Dissertation, Universität Köln, 1989).Google Scholar
2Berndt, J.. Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395 (1989), 132–41.Google Scholar
3Berndt, J.. Real hypersurfaces in quaternionic space forms. J. Reine Angew. Math. 419 (1991), 9–26.Google Scholar
4Chen, B. Y. and Vanhecke, L.. Isometric, holomorphic and symplectic reflections. Geom. Dedicata 29 (1989), 259–77.CrossRefGoogle Scholar
5Chen, B. Y. and Vanhecke, L.. Symplectic reflections in complex space forms. Ann. Global Anal. Geom. 9(1991), 205–10.CrossRefGoogle Scholar
6de Almeida, S. C. and Brito, F.. Minimal hypersurfaces of S4 with constant Gauss–Kronecker curvature. Math. Z. 195 (1987), 99–107.CrossRefGoogle Scholar
7D'Atri, J. E.. Geodesic conformal transformations and symmetric spaces. Kōdai Math. Sem. Rep. 26 (1975), 201–3.Google Scholar
8D'Atri, J. E. and Nickerson, H. K.. Divergence-preserving geodesic symmetries. J. Differential Geom. 3(1969), 467–76.CrossRefGoogle Scholar
9Garcia-Rio, E. and Vanhecke, L.. Geodesic transformations and space forms. Math. J. Toyama Univ. 20(1997), 57–77.Google Scholar
10García-Río, E. and Vanhecke, L.. Geodesic transformations and harmonic spaces. Rend. Circ. Mat. Palermo (to appear).Google Scholar
11Gray, A.. Tubes (Reading: Addison-Wesley, 1990).Google Scholar
12Gray, A. and Vanhecke, L.. The volumes of tubes in a Riemannian manifold. Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), 1–50.Google Scholar
13Gray, A. and Vanhecke, L.. The volumes of tubes about curves in a Riemannian manifold. Proc. London Math. Soc. 44 (1982), 215–43.CrossRefGoogle Scholar
14Kowalski, O. and Vanhecke, L.. G-deformations and some generalizations of H. Weyl's tube theorem. Trans. Amer. Math. Soc. 294 (1986), 799–811.Google Scholar
15Kowalski, O., Prüfer, F. and Vanhecke, L.. D'Attri spaces. In Topics in Geometry: In Memory of Joseph D'Atri, ed. Gindikin, S., Progress in Nonlinear Differential Equations 20, pp. 241–84 (Boston: Birkhäuser, 1996).CrossRefGoogle Scholar
16Smyth, B.. Differential geometry of complex hypersurfaces. Ann. of Math. 85 (1967), 246–66.CrossRefGoogle Scholar
17Tachibana, S.. On Riemannian spaces admitting geodesic conformal transformations. Tensor (N.S.) 25(1972), 323–31.Google Scholar
18Tondeur, Ph. and Vanhecke, L.. Reflections in submanifolds. Geom. Dedicata 28 (1988), 77–85.CrossRefGoogle Scholar
19Tricerri, F. and Vanhecke, L.. Cartan hypersurfaces and reflections. Nihonkai Math. J. 1 (1990), 203–8.Google Scholar
20Vanhecke, L.. Geometry in normal and tubular neighborhoods. Rend. Sem. Fac. Sci. Univ. Cagliari 58(Supp.)(1988), 73–176.Google Scholar
21Vanhecke, L. and Willmore, T. J.. Interaction of tubes and spheres. Math. Ann. 263 (1983), 31–42.CrossRefGoogle Scholar