Abstract
Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set M G. We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of M G. Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify nonnegatively curved fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.
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References
Borel A.: Some remarks about transformation groups on spheres and tori. Bull. Am. Math. Soc. 55, 580–587 (1949)
Borel A.: Le plan projectif des octaves et les spheres comme espaces homogenes. C. R. Acad. Sci. Paris 230, 1378–1381 (1950)
Burago, D., Yuri, B., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics. Vol. 33, American Mathematical Society, USA (2001)
Cao, J., Ge, J.: A simple proof of Perelmans collapsing theorem for 3-manifolds. J. Geom. Anal. (2010). doi:10.1007/s12220-010-9169-5
Cheeger J., Gromoll D.: The splitting theorem for manifolds ofnonnegative Ricci curvature. J. Differ. Geom. 33, 119–128 (1971)
Cheeger J., Gromoll D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96(2) 413–443 (1972)
Fintushel R.: Circle actions on simply connected 4-manifolds. Trans. Am. Math. Soc. 230, 147–171 (1977)
Fintushel R.: Classification of circle actions on 4-manifolds. Trans. Am. Math. Soc. 242, 377–390 (1978)
Fintushel, R., Stern, R.J., Sunukjian, R.: Exotic group actions on simply connected smooth 4-manifolds, arXiv:0902.0963v1 [math.GT] (2009)
Gromov M.: Curvature, diameter and Betti numbers. Comment. Math. Helv. 56(2) 179–195 (1981)
Grove, K.: Geometry of, and via, symmetries. In: Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), Univ. Lecture Ser., vol. 27, pp. 31–53. Am. Math. Soc., Providence, RI (2002)
Grove, K.: Developments around positive sectional curvature. arXiv:0902.4419v1 [math.DG] (2009)
Grove, K., Searle, C.: Positively curved manifolds with maximal symmetry-rank. J. Pure Appl. Algebra 91(1–3), 137–142 (1994)
Grove K., Searle C.: Differential topological restrictions curvature and symmetry. J. Differ. Geom. 47(3), 530–559 (1997)
Grove, K., Wilking, B.: A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry. Preprint, (2011)
Grove K., Wilking B., Ziller W.: Positively curved cohomogeneity one manifolds and 3-Sasakian geometry. J. Differ. Geom. 78(1), 33–111 (2008)
Grove K., Ziller W.: Curvature and symmetry of Milnor spheres. Ann. Math. 152(1), 331–367 (2000)
Hamilton R.S.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)
Hillman J.A.: The algebraic characterization of geometric 4-manifolds, London Mathematical Society Lecture Note Series, vol. 198. Cambridge University Press, Cambridge (1994)
Hoelscher C.A.: Classification of cohomogeneity one manifolds in low dimensions. Pacific J. Math. 246(1), 129–185 (2010)
Hsiang W.Y., Kleiner B: On the topology of positively curved 4-manifolds with symmetry. J. Differ. Geom. 29(3), 615–621 (1989)
Kleiner, B.: Riemannian four-manifolds with nonnegative curvature and continuous symmetry, Ph.D. thesis, University of California, Berkeley, 1990
Kleiner, B., Lott, J.: Notes on Perelman’s papers Preprint (2006) arXiv:math/0605667v2 [math.DG].
Kobayashi, S.: Transformation groups in differential geometry. Classics in Mathematics, Springer, Berlin. Reprint of the 1972 edition (1995)
Montgomery D., Samelson H.: Transformation groups on spheres. Ann. Math. 44, 454–470 (1943)
Morgan J., Tian G.: Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs, vol. 3. Am. Math. Soc., Providence, RI (2007)
Orlik P.: Seifert Manifolds. Lecture Notes in Mathemaics, vol. 291. Springer, New York (1972)
Orlik, P., Raymond, F.: Actions of SO(2) on 3-manifolds. Proceedings of Conference on Transformation Groups, New Orleans, LA (1967), pp. 297–318. Springer, New York (1968)
Orlik P., Raymond F.: Actions of the torus on 4-manifolds. I. Trans. Am. Math. Soc. 152, 531–559 (1970)
Pao P.S.: Nonlinear circle actions on the 4-sphere and twisting spun knots. Topology 17(3), 291–296 (1978)
Perelman, G.: Alexandrov spaces with curvature bounded below, ii, Preprint
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint (2002), arXiv:math/0211159v1 [math.DG]
Perelman, G.: Ricci flow with surgery on three-manifolds. Preprint (2003), arXiv:math/0303109v1 [math.DG]
Poncet J.: Groupes de lie compacts de transformations de l’espaces euclidean et les spheres comme espaces homogenes. Comment. Math. Helv. 33, 109–120 (1959)
Raymond F.: Classification of the actions of the circle on 3-manifolds. Trans. Am. Math. Soc. 131, 51–78 (1968)
Searle C., Yang D.: On the topology of non-negatively curved simply connected 4-manifolds with continuous symmetry. Duke Math. J. 74(2), 547–556 (1994)
Seifert H.: Topologie Dreidimensionaler Gefaserter Räume. Acta Math. 60(1), 147–238 (1933)
Shioya T., Yamaguchi T.: Collapsing three-manifolds under a lower curvature bound. J. Differ. Geom. 56(1), 1–66 (2000)
Verdiani L.: Cohomogeneity one manifolds of even dimension with strictly positive sectional curvature. J. Differ. Geom. 68(1), 31–72 (2004)
Wilking B.: Positively curved manifolds with symmetry. Ann. Math. 163(2), 607–668 (2006)
Wilking, B.: Nonnegatively and positively curved manifolds. Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, pp. 25–62. Somerville, MA (2007)
Ziller, W.: Examples of Riemannian manifolds with non-negative sectional curvature. Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, pp. 63–102. Somerville, MA (2007)
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Galaz-Garcia, F. Nonnegatively curved fixed point homogeneous manifolds in low dimensions. Geom Dedicata 157, 367–396 (2012). https://doi.org/10.1007/s10711-011-9615-y
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DOI: https://doi.org/10.1007/s10711-011-9615-y