POSITIVELY CURVED MANIFOLDS WITH FIXED POINT COHOMOGENEITY ONE

Title & Authors
POSITIVELY CURVED MANIFOLDS WITH FIXED POINT COHOMOGENEITY ONE
KIM CHANG-WAN;

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
positive curvature;transformation groups;fixed point cohomogeneity;
Language
English
Cited by
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

2.
A. Back and W. -Y. Hsiang, Equivariant geometry and Kervaire spheres, Trans. Amer. Math. Soc. 304 (1987), 207-227

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

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

10.
F. Fang and X. Rong, Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank, Math. Ann. 332 (2005), 81-101

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

13.
G. Grove and C. -W. Kim, Positively curved manifolds with low fixed point cohomogeneity, J. Differential Geom. 67 (2004), 1-33

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

15.
K. Grove and C. Seale, Differential topological restrictions by curvature, and symmetry, J. Differential Geom. 47 (1997), 530-559

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

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

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

24.
L. Verdiani, Cohomogeneity one Riemannian manifolds of even dimension with strictly positive sectional curvature, J. Differential Geom. 68 (2004), 31-72

25.
N. R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 (1972), 277-295

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

28.
B. Wilking, Positively curved manifolds with symmetry, Ann. of Math. 163 (2006), 607-668