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Note on a theorem of Gromoll-Grove

  • Grant Cairns (a1) and Richard H. Escobales (a2)

Abstract

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.

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References

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[1]Basmajian, A. and Walschap, G., ‘Metric flows in space forms of nonpositive curvature’, Proc. Amer. Math. Soc. 123 (1995), 31773181.
[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).
[3]Domínguez, D., ‘Finiteness and tenseness theorems for Riemannian foliations’, (preprint).
[4]Escobales, R. Jr, and Parker, P., ‘Geometric consequences of the normal curvature cohomology class in umbilic foliations’, Indiana Univ. Math. J. 37 (1988), 389408.
[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. 165168.
[6]Milnor, J., ‘Curvatures of left invariant metrics on Lie groups’, Adv. in Math. 21 (1976), 293329.
[7]Molino, P., Riemannian foliations (Birkhauser, Boston, Basel, 1988).
[8]Nagano, T., ‘On fibred Riemann manifolds’, Sci. Papers Gen. Ed. Univ. Tokyo 10 (1960), 1727.
[9]O'Neill, B., ‘The fundamental equations of a submersion’, Michigan Math. J. 13 (1966), 459469.
[10]Reinhart, B., ‘Foliated manifolds with bundle-like metrics’, Ann. of Math. 69 (1959), 119131.
[11]Tondeur, P., Foliations on Riemannian Manifolds (Springer-Verlag, Berlin, Heidelberg, New York, 1988).
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