Geometry & Topology

Nonnegatively curved $5$–manifolds with almost maximal symmetry rank

Abstract

We show that a closed, simply connected, nonnegatively curved $5$–manifold admitting an effective, isometric $T2$ action is diffeomorphic to one of $S5,S3×S2$, $S3×̃S2$ or the Wu manifold $SU(3)∕SO(3)$.

Article information

Source
Geom. Topol., Volume 18, Number 3 (2014), 1397-1435.

Dates
Revised: 8 November 2013
Accepted: 14 December 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732798

Digital Object Identifier
doi:10.2140/gt.2014.18.1397

Mathematical Reviews number (MathSciNet)
MR3228455

Zentralblatt MATH identifier
1294.53038

Citation

Galaz-Garcia, Fernando; Searle, Catherine. Nonnegatively curved $5$–manifolds with almost maximal symmetry rank. Geom. Topol. 18 (2014), no. 3, 1397--1435. doi:10.2140/gt.2014.18.1397. https://projecteuclid.org/euclid.gt/1513732798

References

• D Barden, Simply connected $5$–manifolds, Ann. of Math. 82 (1965) 365–385
• G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press, New York (1972)
• D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, Amer. Math. Soc. (2001)
• Y Burago, M Gromov, G Perelman, AD Aleksandrov's spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3–51, 222 In Russian; translated in Russian Math. Surveys 47 (1992) 1–58
• T Chang, T Skjelbred, Group actions on Poincaré duality spaces, Bull. Amer. Math. Soc. 78 (1972) 1024–1026
• J Cheeger, D G Ebin, Comparison theorems in Riemannian geometry, AMS Chelsea Publishing (2008)
• J DeVito, The classification of simply connected biquotients of dimension at most $7$ and $3$ new examples of almost positively curved manifolds, PhD thesis, University of Pennsylvania (2011) Available at \setbox0\makeatletter\@url http://search.proquest.com//docview/878684574 {\unhbox0
• R Diestel, Graph theory, 3rd edition, Graduate Texts in Mathematics 173, Springer, Berlin (2005)
• J Dinkelbach, B Leeb, Equivariant Ricci flow with surgery and applications to finite group actions on geometric $3$–manifolds, Geom. Topol. 13 (2009) 1129–1173
• H Duan, C Liang, Circle bundles over $4$–manifolds, Arch. Math. (Basel) 85 (2005) 278–282
• R Fintushel, Circle actions on simply connected $4$–manifolds, Trans. Amer. Math. Soc. 230 (1977) 147–171
• R Fintushel, Classification of circle actions on $4$–manifolds, Trans. Amer. Math. Soc. 242 (1978) 377–390
• F Galaz-Garcia, Nonnegatively curved fixed point homogeneous manifolds in low dimensions, Geom. Dedicata 157 (2012) 367–396
• F Galaz-Garcia, M Kerin, Cohomogeneity–two torus actions on nonnegatively curved manifolds of low dimension, Math. Z. 276 (2014) 133–152
• F Galaz-Garcia, C Searle, Low-dimensional manifolds with nonnegative curvature and maximal symmetry rank, Proc. Amer. Math. Soc. 139 (2011) 2559–2564
• F Galaz-Garcia, W Spindeler, Nonnegatively curved fixed point homogeneous $5$–manifolds, Ann. Global Anal. Geom. 41 (2012) 253–263
• F Galaz-Garcia, W Spindeler, Erratum to: Nonnegatively curved fixed point homogeneous $5$–manifolds, Ann. Global Anal. Geom. 45 (2014) 151–153
• J C Gómez-Larrañaga, F González-Acuña, W Heil, $S\sp 2$– and $P\sp 2$–category of manifolds, Topology Appl. 159 (2012) 1052–1058
• J L Gross, J Yellen (editors), Handbook of graph theory, CRC Press, Boca Raton, FL (2004)
• K Grove, Geometry of, and via, symmetries, from “Conformal, Riemannian and Lagrangian geometry”, Univ. Lecture Ser. 27, Amer. Math. Soc. (2002) 31–53
• K Grove, Developments around positive sectional curvature, from “Geometry, analysis, and algebraic geometry” (H-D Cao, S-T Yau, editors), Surv. Differ. Geom. 13, Int. Press (2009) 117–133
• K Grove, C Searle, Positively curved manifolds with maximal symmetry rank, J. Pure Appl. Algebra 91 (1994) 137–142
• K Grove, C Searle, Differential topological restrictions curvature and symmetry, J. Differential Geom. 47 (1997) 530–559
• K Grove, B Wilking, A knot characterization and $1$–connected nonnegatively curved $4$–manifolds with circle symmetry
• B A Kleiner, Riemannian four-manifolds with nonnegative curvature and continuous symmetry, PhD thesis, University of California, Berkeley (1990) Available at \setbox0\makeatletter\@url http://search.proquest.com//docview/303876774 {\unhbox0
• D Montgomery, C T Yang, Groups on $S\sp{n}$ with principal orbits of dimension $n-3$, Illinois J. Math. 4 (1960) 507–517
• P S Mostert, On a compact Lie group acting on a manifold, Ann. of Math. 65 (1957) 447–455
• W D Neumann, $3$–dimensional $G$–manifolds with $2$–dimensional orbits, from “Proc. Conf. Transformation Groups”, Springer, New York (1968) 220–222
• P Orlik, Seifert manifolds, Lecture Notes in Math. 291, Springer, Berlin (1972)
• P Orlik, F Raymond, Actions of ${\rm SO}(2)$ on $3$–manifolds, from “Proc. Conf. Transformation Groups”, Springer, New York (1968) 297–318
• P S Pao, Nonlinear circle actions on the $4$–sphere and twisting spun knots, Topology 17 (1978) 291–296
• G P Paternain, J Petean, Minimal entropy and collapsing with curvature bounded from below, Invent. Math. 151 (2003) 415–450
• A V Pavlov, Five-dimensional biquotients of Lie groups, Sibirsk. Mat. Zh. 45 (2004) 1323–1328
• G Perelman, The entropy formula for the Ricci flow and its geometric applications
• G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• G Perelman, Ricci flow with surgery on three-manifolds
• G Perelman, Alexandrov's spaces with curvatures bounded from below, II, preprint (1991)
• F Raymond, Classification of the actions of the circle on $3$–manifolds, Trans. Amer. Math. Soc. 131 (1968) 51–78
• X Rong, Positively curved manifolds with almost maximal symmetry rank, from “Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II”, 95 (2002) 157–182
• P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401–487
• C Searle, D Yang, On the topology of nonnegatively curved simply connected $4$–manifolds with continuous symmetry, Duke Math. J. 74 (1994) 547–556
• S Smale, On the structure of $5$–manifolds, Ann. of Math. 75 (1962) 38–46
• W Spindeler, Fixpunkthomogene $S\sp 1$–Wirkungen auf $5$–Mannigfaltigkeiten nichtnegativer Krümmung (2009)
• B Totaro, Cheeger manifolds and the classification of biquotients, J. Differential Geom. 61 (2002) 397–451
• B Wilking, Torus actions on manifolds of positive sectional curvature, Acta Math. 191 (2003) 259–297
• B Wilking, Nonnegatively and positively curved manifolds, from “Metric and comparison geometry” (J Cheeger, K Grove, editors), Surv. Differ. Geom. 11, Int. Press (2007) 25–62
• B Wilking, Group actions on nonnegatively and positively curved manifolds, Lecture notes, Münster (2010)
• W Ziller, Examples of Riemannian manifolds with nonnegative sectional curvature, from “Metric and comparison geometry” (J Cheeger, K Grove, editors), Surv. Differ. Geom. 11, Int. Press (2007) 63–102