DOI QR코드

DOI QR Code

POSITIVELY CURVED MANIFOLDS WITH FIXED POINT COHOMOGENEITY ONE

  • Published : 2006.01.01

Abstract

Any simply connected fixed point cohomogeneity one riemannian manifold with positive sectional curvature is diffeomorphic to one of the compact rank one symmetric spaces.

Keywords

References

  1. S. Aloff and N. R. Wallach, An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93- 97 https://doi.org/10.1090/S0002-9904-1975-13649-4
  2. A. Back and W. -Y. Hsiang, Equivariant geometry and Kervaire spheres, Trans. Amer. Math. Soc. 304 (1987), 207-227 https://doi.org/10.2307/2000711
  3. L. Berard-Bergery, Les varietes riemanniennes homogµenes simplement connexes de dimension impaire a courbure strictement positive, J. Math. Pures Appl. 55 (1976), 47-67
  4. M. Berger, Les varietes riemanniennes homogµenes normales simplement connexes a courbure strictement positive, Ann. Scuola Norm. Sup. Pisa 15 (1961), 179-246
  5. G. E. Bredon, On the structure of orbit spaces of generalized manifolds, Trans. Amer. Math. Soc. 100 (1961), 162-196 https://doi.org/10.2307/1993358
  6. G. E. Bredon, Introduction to compact transformation groups, Academic Press, 1972
  7. Y. Burago, M. Gromov, and G. Perelman, A. D. Aleksandrov spaces with curva- tures bounded below, Uspekhi Mat. Nauk 47 (1992), 3-51
  8. Y. Burago, M. Gromov, and G. Perelman, A. D. Aleksandrov spaces with curva- tures bounded below, translation in Russian Math. Surveys 47 (1992), 1-58
  9. J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnega- tive curvature, Ann. of Math. 96 (1972), 413-443 https://doi.org/10.2307/1970819
  10. F. Fang and X. Rong, Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank, Math. Ann. 332 (2005), 81-101 https://doi.org/10.1007/s00208-004-0618-y
  11. K. Grove, Geometry of, and via, symmetries, Univ. Lecture Ser., Amer. Math. Soc., Providence, RT 27 (2002), 31-53
  12. K. Grove and S. Halperin, Dupin hypersurfaces, group actions and the double mapping cylinder, J. Differential Geom. 26 (1987), 429-459 https://doi.org/10.4310/jdg/1214441486
  13. G. Grove and C. -W. Kim, Positively curved manifolds with low fixed point cohomogeneity, J. Differential Geom. 67 (2004), 1-33 https://doi.org/10.4310/jdg/1099587728
  14. K. Grove and S. Markvosen, New extremal problems for the Riemannian recognition program via Alexandrov geometry, J. Amer. Math. Soc. 8 (1995), 1-28 https://doi.org/10.2307/2152882
  15. K. Grove and C. Seale, Differential topological restrictions by curvature, and symmetry, J. Differential Geom. 47 (1997), 530-559 https://doi.org/10.4310/jdg/1214460549
  16. K. Grove, B. Wilking, and W. Ziller, Positively curved cohomogeneity one manifolds and 3-Sasakian geometry, http://kr.arxiv.org/abs/math.DG/0511464
  17. W. -Y. Hsiang and H. B. Lawson, Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1-38 https://doi.org/10.4310/jdg/1214429775
  18. C. -W. Kim, Positively curved manifolds with orbits uniform dimension, preprint
  19. B. Kleiner, Riemannian 4-manifolds with nonnegative curvature and continuous symmetry, Ph. D. thesis, U. C. Berkeley, 1990
  20. H. F. Munzner, Isoparametrische hyperachen in spharen, Math. Ann. 251 (1980), 57-71 https://doi.org/10.1007/BF01420281
  21. H. F. Munzner, Isoparametrische hyperachen in spharen, Uber die zerlegung der Sphare in Ballbundel. II, 256 (1981), 427-440
  22. G. Perelman, Alexandrov spaces with curvature bounded below, ii, preprint
  23. F. Uchida, An orthogonal transformation group of the 8(k -1)-sphere, J. Differential Geom. 15 (1980), 569-574 https://doi.org/10.4310/jdg/1214435844
  24. L. Verdiani, Cohomogeneity one Riemannian manifolds of even dimension with strictly positive sectional curvature, J. Differential Geom. 68 (2004), 31-72 https://doi.org/10.4310/jdg/1102536709
  25. N. R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 (1972), 277-295 https://doi.org/10.2307/1970789
  26. B. Wilking, The normal homogeneous space ($(SU(3){\times}SO(3))/U^{\bullet}(2)$ has positive sectional curvature, Proc. Amer. Math. Soc. 127 (1999), 1191-1194
  27. B. Wilking, Torus actions on manifolds of positive sectional curvature, Acta Math. 191 (2003), 259-297 https://doi.org/10.1007/BF02392966
  28. B. Wilking, Positively curved manifolds with symmetry, Ann. of Math. 163 (2006), 607-668 https://doi.org/10.4007/annals.2006.163.607