An example relevant to curvature pinching theorems for Riemannian foliations

Hebda, James J.

doi:10.1090/S0002-9939-1992-1086327-7

An example relevant to curvature pinching theorems for Riemannian foliations
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by James J. Hebda PDF

Proc. Amer. Math. Soc. 114 (1992), 195-199 Request permission

Abstract:

We describe a one-parameter family of codimension-$2$ Riemannian foliations of ${S^3}$ which are not transversely homogeneous. The example shows that the comparison theorem for Riemannian foliations of Kamber-Ruh-Tondeur [4] fails without an additional hypothesis.

References

Robert A. Blumenthal, Transversely homogeneous foliations, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, vii, 143–158 (English, with French summary). MR 558593
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
F. W. Kamber, E. A. Ruh, and Ph. Tondeur, Almost transversally symmetric foliations, Differential geometry, Peñíscola 1985, Lecture Notes in Math., vol. 1209, Springer, Berlin, 1986, pp. 184–189. MR 863756, DOI 10.1007/BFb0076631
Franz W. Kamber, Ernst A. Ruh, and Philippe Tondeur, Comparing Riemannian foliations with transversally symmetric foliations, J. Differential Geom. 27 (1988), no. 3, 461–475. MR 940114
Philippe Tondeur, The mean curvature of Riemannian foliations, Feuilletages riemanniens, quantification géométrique et mécanique (Lyon, 1986) Travaux en Cours, vol. 26, Hermann, Paris, 1988, pp. 41–52. MR 938306, DOI 10.1007/978-1-4613-8780-0