Nonnegatively curved fixed point homogeneous 5-manifolds
- First Online:
Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.
KeywordsNonnegative curvature Circle action 5-manifold Fixed point homogeneous
Mathematics Subject Classification (2000)53C20
Unable to display preview. Download preview PDF.
- 3.Bredon G.E.: Introduction to Compact Transformation Groups, Pure and Applied Mathematics, vol. 46. Academic Press, New York, London (1972)Google Scholar
- 4.Burago D., Yuri B., Ivanov S.: A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001)Google Scholar
- 6.Galaz-Garcia, F.: Nonnegatively curved fixed point homogeneous manifolds in low dimensions. Geom. Dedicata, to appear. arXiv:0911.1254v1 [math.DG]Google Scholar
- 7.Galaz-Garcia, F., Searle, C.: Nonnegatively curved 5-manifolds with almost maximal symmetry rank. Preprint (2011) (see arXiv:0906.3870v1 [math.DG])Google Scholar
- 8.Grove, K.: Critical point theory for distance functions. Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), Proceedings of the Symposium in Pure Mathematics, vol. 54, Part 3, pp. 357–385. American Mathematical Society, Providence, RI (1993)Google Scholar
- 9.Grove, K.: Geometry of, and via, symmetries. Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), University Lecture Series, vol. 27, pp. 31–53. American Mathematical Society, Providence, RI (2002)Google Scholar
- 17.Orlik, P., Raymond, F.: Actions of SO(2) on 3-manifolds, Proceedings of the Conference on Transformation Groups (New Orleans, La., 1967), pp. 297–318. Springer, New York (1968)Google Scholar
- 18.Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint (2002), arXiv:math/0211159v1 [math.DG].Google Scholar
- 19.Perelman, G.: Ricci flow with surgery on three-manifolds. Preprint (2003), arXiv:math/0303109v1 [math.DG]Google Scholar
- 29.Wilking, B.: Nonnegatively and positively curved manifolds. Surveys in differential geometry. vol. XI, Surveys in differential Geometry, vol. 11, pp. 25–62. International Press, Somerville, MA (2007)Google Scholar