Totally geodesic foliations on $3$-manifolds
HTML articles powered by AMS MathViewer
- by David L. Johnson and Lee B. Whitt PDF
- Proc. Amer. Math. Soc. 76 (1979), 355-357 Request permission
Abstract:
If M is a compact 3-manifold, it is known that M can be foliated by 2-manifolds. Topological obstructions are given to the geodesibility of such a foliation $\mathcal {F}$; that is, to the existence of a Riemannian metric on M making each leaf a totally geodesic submanifold. For example, ${\pi _1}(M)$ must be infinite, and hence the Reeb foliation of ${S^3}$ is not geodesible.References
-
H. Gluck, Open letter on geodesible flows.
- David L. Johnson, Kähler submersions and holomorphic connections, J. Differential Geometry 15 (1980), no. 1, 71–79 (1981). MR 602440
- H. Blaine Lawson Jr., Foliations, Bull. Amer. Math. Soc. 80 (1974), 369–418. MR 343289, DOI 10.1090/S0002-9904-1974-13432-4
- S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248–278 (Russian). MR 0200938
- Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
- W. P. Thurston, Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268. MR 425985, DOI 10.2307/1971047
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 76 (1979), 355-357
- MSC: Primary 57R30
- DOI: https://doi.org/10.1090/S0002-9939-1979-0537106-1
- MathSciNet review: 537106