Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

COHOMOGENEITY ONE RIEMANNIAN MANIFOLDS OF CONSTANT POSITIVE CURVATURE

  • Published : 2007.07.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we study non-simply connected Riemannian manifolds of constant positive curvature which have an orbit of codimension one under the action of a connected closed Lie subgroup of isometries. When the action is reducible we characterize the orbits explicitly. We also prove that in some cases the manifold is homogeneous.

Keywords

References

  1. A. V. Alekseevsky and D. V. Alekseevsky, G-manifolds with one-dimensional orbit space, Lie groups, their discrete subgroups, and invariant theory, 1-31, Adv. Soviet Math. 8, Amer. Math. Soc., Providence, RI, 1992
  2. A. V. Alekseevsky and D. V. Alekseevsky, Riemannian G-manifold with one-dimensional orbit space, Ann. Global Anal. Geom. 11 (1993), no. 3, 197-211
  3. H. Borel, Some remarks about Lie groups transitive on spheres and tori, Bull. Amer. Math. Soc. 55 (1949). 580-587 https://doi.org/10.1090/S0002-9904-1949-09251-0
  4. J. Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), no. 1, 125-137 https://doi.org/10.2307/2000430
  5. A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354 (2002), no. 2, 571-612 https://doi.org/10.1090/S0002-9947-01-02803-3
  6. F. Podestµa and L. Verdiani, Positively curved 7-dimensional manifolds, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 497-504 https://doi.org/10.1093/qjmath/50.200.497
  7. C. Searle, Cohomogeneity and positive curvature in low dimensions, Math. Z. 214 (1993), no. 3, 491-498 https://doi.org/10.1007/BF02572419
  8. E. Straume, Compact connected Lie transformation groups on spheres with low cohomogeneity. I., Mem. Amer. Math. Soc. 119 (1996), no. 569
  9. J. A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney 1967