Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T06:06:21.477Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant submanifolds in flow geometry

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  09 April 2009

J. C. González-Dávila ,
M. C. González-Dávila  and
L. Vanhecke
J. C. González-Dávila
Affiliation:
Department o de Matemática Fundamental Sección de Geometría y Topología Universidad de La LagunaLa Laguna, Spain
M. C. González-Dávila
Affiliation:
Department of Mathematics Katholieke Universiteit LeuvenCelestijnenlaan 200 B B-3001 Leuven, Belgium
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We begin a study of invariant isometric immersions into Riemannian manifolds (M, g) equipped with a Riemannian flow generated by a unit Killing vector field ξ. We focus our attention on those (M, g) where ξ is complete and such that the reflections with respect to the flow lines are global isometries (that is, (M, g) is a Killing-transversally symmetric space) and on the subclass of normal flow space forms. General results are derived and several examples are provided.

Keywords

Isometricnormal and contact flowsKilling-transversally symmetric spacesnormal fow space formsinvariant immersions and embeddings

MSC classification

Secondary: 53B25: Local submanifolds 53C12: Foliations (differential geometric aspects) 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C35: Symmetric spaces 53C40: Global submanifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 3 , June 1997 , pp. 290 - 314
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Besse, A. L., Einstein manifolds, Ergeb. Math. Grenzgeb. 3. Folge 10 (Springer, Berlin, 1987).CrossRefGoogle Scholar
[2]Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509 (Springer, Berlin, 1976).CrossRefGoogle Scholar
[3]Boothby, W. M. and Wang, H. C., ‘On contact manifolds’, Ann. of Math. 68 (1958), 721–734.CrossRefGoogle Scholar
[4]Calabi, E., ‘Isometric embedding of complex manifolds’, Ann. of Math. 58 (1953), 1–23.CrossRefGoogle Scholar
[5]González-Dávila, J. C., González-Dávila, M. C. and Vanhecke, L., ‘Reflections and isometric flows’, Kyungpook Math. J. 35 (1995), 113–144.Google Scholar
[6]González-Dávila, J. C., González-Dávila, M. C. and Vanhecke, L., ‘Classification of Killing-transversally symmetric spaces’, Tsukuba J. Math. 20 (1996), 321–347.CrossRefGoogle Scholar
[7]González-Dávila, J. C., González-Dávila, M. C. and Vanhecke, L., ‘Normal flow forms and their classification’, Publ. Math. Debrecen 48 (1996), 151–173.CrossRefGoogle Scholar
[8]González-Dávila, M. C., ‘KTS-spaces and natural reductivity’, Nihonkai Math. J. 5 (1994), 115–129.Google Scholar
[9]Jiménez, J. A. and Kowalski, O., ‘The classification of φ-symmetric Sasakian manifolds’, Monatsh. Math. 115 (1993), 83–98.CrossRefGoogle Scholar
[10]Kobayashi, S., ‘Principal fibre bundles with the 1-dimensional toroidal group’, Tôhoku Math. J. 8 (1956), 29–45.CrossRefGoogle Scholar
[11]Kon, M., ‘On invariant submanifolds in a Sasakian manifold of constant φ-sectional curvature’, TRU Math. 10 (1974), 1–9.Google Scholar
[12]Kon, M., ‘Invariant submanifolds in Sasakian manifolds’, Math. Ann. 219 (1976), 277–290.CrossRefGoogle Scholar
[13]Nakagawa, H. and Takagi, R., ‘On locally symmetric Kaehler submanifolds in a complex projective space’, J. Math. Soc. Japan 28 (1976), 638–667.CrossRefGoogle Scholar
[14]O'Neill, B., ‘The fundamental equation of a submersion’, Michigan Math. J. 13 (1966), 459–469.CrossRefGoogle Scholar
[15]Ogiue, K., ‘On fiberings of almost contact manifolds’, Kōdai Math. Sem. Rep. 17 (1965), 53–62.Google Scholar
[16]Ogiue, K., ‘Differential geometry of algebraic manifolds’, in: Differential geometry (in honor of Yano, K..) (Kinokuniya, Tokyo, 1972) pp. 355–372.Google Scholar
[17]Ogiue, K., ‘Positively curved complex hypersurfaces immersed in a complex projective space’, Tôhoku Math. J. 24 (1972), 51–54.CrossRefGoogle Scholar
[18]Ogiue, K., ‘Positively curved complex submanifolds immersed in a complex projective space’ J. Differential Geom. 7 (1972), 603–606.CrossRefGoogle Scholar
[19]Ogiue, K., ‘On Kaehler immersions’, Canad. J. Math. 24 (1972), 1178–1182.CrossRefGoogle Scholar
[20]Ogiue, K., ‘n-dimensional complex space forms immersed in [n + (n(n + l)/2)]-dimensional complex space forms’, J. Math. Soc. Japan 24 (1972), 518–526.CrossRefGoogle Scholar
[21]Ogiue, K., ‘Differential geometry of Kaehler submanifolds’, Adv. Math. 13 (1974), 73–114.CrossRefGoogle Scholar
[22]Palais, R. S., A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. 22 (Amer. Math. Soc., Providence, 1957).CrossRefGoogle Scholar
[23]Reinhart, B. L., Differential geometry of foliations, Ergeb. Math. Grenzgeb. 99 (Springer, Berlin, 1983).CrossRefGoogle Scholar
[24]Smyth, B., ‘Differential geometry of complex hypersurfaces’, Ann. of Math. 85 (1967), 246–266.CrossRefGoogle Scholar
[25]Tondeur, Ph., Foliations on Riemannian manifolds, Universitext (Springer, Berlin, 1988).CrossRefGoogle Scholar
[26]Tondeur, Ph. and Vanhecke, L., ‘Transversally symmetric Riemannian foliations’, Tôhoku Math. J. 42 (1990), 307–317.CrossRefGoogle Scholar
[27]Wang, H. C., ‘Closed manifolds with homogeneous complex structures’, Amer. J. Math. 76 (1954), 1–32.CrossRefGoogle Scholar
[28]Warner, F. W., Foundations of differentiable manifolds and Lie groups (Scott, Foresman, Glenview, 1971).Google Scholar
[29]Yano, K. and Kon, M., Structures on manifolds, Ser. Pure Math. 3 (World Scientific Publ. Co., Singapore, 1984).Google Scholar