Some properties of closed 1-forms on a special Riemannian manifold

Tsagas, Gr.

doi:10.1090/S0002-9939-1981-0589147-5

Some properties of closed $1$-forms on a special Riemannian manifold
HTML articles powered by AMS MathViewer

by Gr. Tsagas

Proc. Amer. Math. Soc. 81 (1981), 104-106

DOI: https://doi.org/10.1090/S0002-9939-1981-0589147-5

PDF | Request permission

Abstract:

Let $M$ be a compact Riemannian manifold whose sectional curvature is strictly negative; then every closed $1$-form on $M$ has a singularity.

References

R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
William P. Byers, On a theorem of Preissmann, Proc. Amer. Math. Soc. 24 (1970), 50–51. MR 251665, DOI 10.1090/S0002-9939-1970-0251665-X
F. T. Farrell, The obstruction to fibering a manifold over a circle, Indiana Univ. Math. J. 21 (1971/72), 315–346. MR 290397, DOI 10.1512/iumj.1971.21.21024
Detlef Gromoll and Joseph A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971), 545–552. MR 281122, DOI 10.1090/S0002-9904-1971-12747-7
Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR 232311
John Milnor, Growth of finitely generated solvable groups, J. Differential Geometry 2 (1968), 447–449. MR 244899
Grigorios Tsagas, On the singularities of harmonic $1$-forms on a Riemannian manifold, K\B{o}dai Math. Sem. Rep. 26 (1975), no. 4, 459–466. MR 394506
D. Tischler, On fibering certain foliated manifolds over $S^{1}$, Topology 9 (1970), 153–154. MR 256413, DOI 10.1016/0040-9383(70)90037-6
Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688
Shing-tung Yau, On the fundamental group of compact manifolds of non-positive curvature, Ann. of Math. (2) 93 (1971), 579–585. MR 283726, DOI 10.2307/1970888