Volumes of flows
HTML articles powered by AMS MathViewer
- by David L. Johnson
- Proc. Amer. Math. Soc. 104 (1988), 923-931
- DOI: https://doi.org/10.1090/S0002-9939-1988-0964875-4
- PDF | Request permission
Abstract:
If $F$ is an oriented nonsingular flow on a Riemannian manifold $M$, the volume of $F$ is defined as the $n$-dimensional measure of the unit vector field tangent to $F$, as a section of ${T_ * }\left ( M \right )$ with the induced metric. It is shown that, for any metric of the two-dimensional torus, and for any homotopy class of flows on the torus, there is a unique smooth flow of minimal volume within the homotopy class. It has been shown that the Hopf foliation on the round threesphere absolutely minimizes the volume of flows on ${S^3}$. In higher dimensions this is not the case; the Hopf fibrations are not even local minima of the volume functional for flows on the round five-sphere. It is not known whether a volume-minimizing flow on ${S^5}$ exists.References
- Melvin S. Berger, Nonlinearity and functional analysis, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis. MR 0488101
- I. Chavel and E. Feldman, The first eigenvalue of the Laplacian on manifolds of non-negative curvature, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 351–353. MR 0378002
- Richard H. Escobales Jr., Riemannian foliations of the rank one symmetric spaces, Proc. Amer. Math. Soc. 95 (1985), no. 3, 495–498. MR 806095, DOI 10.1090/S0002-9939-1985-0806095-7
- J. Girbau, A. Haefliger, and D. Sundararaman, On deformations of transversely holomorphic foliations, J. Reine Angew. Math. 345 (1983), 122–147. MR 717890, DOI 10.1515/crll.1983.345.122 H. Gluck, Can space be filled by geodesies, and if so, how?, manuscript.
- Herman Gluck, Dynamical behavior of geodesic fields, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 190–215. MR 591184
- Herman Gluck and Wolfgang Ziller, On the volume of a unit vector field on the three-sphere, Comment. Math. Helv. 61 (1986), no. 2, 177–192. MR 856085, DOI 10.1007/BF02621910
- James L. Heitsch, A cohomology for foliated manifolds, Comment. Math. Helv. 50 (1975), 197–218. MR 372877, DOI 10.1007/BF02565746
- David L. Johnson, Deformations of totally geodesic foliations, Geometry and topology (Athens, Ga., 1985) Lecture Notes in Pure and Appl. Math., vol. 105, Dekker, New York, 1987, pp. 167–178. MR 873293 —, Families of foliations (to appear).
- David L. Johnson and Lee B. Whitt, Totally geodesic foliations, J. Differential Geometry 15 (1980), no. 2, 225–235 (1981). MR 614368
- L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 267–302. MR 609039, DOI 10.1090/S0273-0979-1981-14888-6
- Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
- Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
- Shigeo Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2) 10 (1958), 338–354. MR 112152, DOI 10.2748/tmj/1178244668
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 923-931
- MSC: Primary 53C20; Secondary 58F17
- DOI: https://doi.org/10.1090/S0002-9939-1988-0964875-4
- MathSciNet review: 964875