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Note on a theorem of Gromoll-Grove

Published online by Cambridge University Press:  17 April 2009

Grant Cairns  and
Richard H. Escobales Jr
Grant Cairns
Affiliation:
School of Mathematics, La Trobe University, Melbourne Vic. 3083, Australia, e-mail: [email protected]
Richard H. Escobales Jr
Affiliation:
Department of Mathematics, Canisius College, Buffalo, NY 14208, United States of America, e-mail: [email protected]
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Abstract

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D. Gromoll and K. Grove showed that metric flows on constant curvature spaces are either flat or locally spanned by Killing vector fields. We generalise this result to certain flows on manifolds of variable curvature.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 1 , February 1997 , pp. 1 - 5
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Basmajian, A. and Walschap, G., ‘Metric flows in space forms of nonpositive curvature’, Proc. Amer. Math. Soc. 123 (1995), 3177–3181.CrossRefGoogle Scholar
[2]Cairns, G. and Escobales, R. Jr, ‘Further geometry of the mean curvature one-form and the normal-plane field one-form on a foliated Reimannian manifold’, J. Aust. Math. Soc. (to appear).Google Scholar
[3]Domínguez, D., ‘Finiteness and tenseness theorems for Riemannian foliations’, (preprint).Google Scholar
[4]Escobales, R. Jr, and Parker, P., ‘Geometric consequences of the normal curvature cohomology class in umbilic foliations’, Indiana Univ. Math. J. 37 (1988), 389–408.Google Scholar
[5]Gromoll, D. and Grove, K., ‘One-dimensional metric foliations in constant curvature spaces’, in Differential geometry and complex analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1985), pp. 165–168.CrossRefGoogle Scholar
[6]Milnor, J., ‘Curvatures of left invariant metrics on Lie groups’, Adv. in Math. 21 (1976), 293–329.CrossRefGoogle Scholar
[7]Molino, P., Riemannian foliations (Birkhauser, Boston, Basel, 1988).CrossRefGoogle Scholar
[8]Nagano, T., ‘On fibred Riemann manifolds’, Sci. Papers Gen. Ed. Univ. Tokyo 10 (1960), 17–27.Google Scholar
[9]O'Neill, B., ‘The fundamental equations of a submersion’, Michigan Math. J. 13 (1966), 459–469.Google Scholar
[10]Reinhart, B., ‘Foliated manifolds with bundle-like metrics’, Ann. of Math. 69 (1959), 119–131.CrossRefGoogle Scholar
[11]Tondeur, P., Foliations on Riemannian Manifolds (Springer-Verlag, Berlin, Heidelberg, New York, 1988).CrossRefGoogle Scholar