Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions
HTML articles powered by AMS MathViewer
- by Diego Corro and Fernando Galaz-García PDF
- Proc. Amer. Math. Soc. 148 (2020), 3087-3097 Request permission
Abstract:
We show that for each $n\geqslant 1$, there exist infinitely many spin and non-spin diffeomorphism types of closed, smooth, simply-connected $(n+4)$-manifolds with a smooth, effective action of a torus $T^{n+2}$ and a metric of positive Ricci curvature invariant under a $T^{n}$-subgroup of $T^{n+2}$. As an application, we show that every closed, smooth, simply-connected $5$- and $6$-manifold admitting a smooth, effective torus action of cohomogeneity two supports metrics with positive Ricci curvature invariant under a circle or $T^2$-action, respectively.References
- D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR 184241, DOI 10.2307/1970702
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- Charles P. Boyer and Krzysztof Galicki, Rational homology 5-spheres with positive Ricci curvature, Math. Res. Lett. 9 (2002), no. 4, 521–528. MR 1928872, DOI 10.4310/MRL.2002.v9.n4.a12
- Charles P. Boyer and Krzysztof Galicki, Erratum and addendum for: “Rational homology 5-spheres with positive Ricci curvature” [Math. Res. Lett. 9 (2002), no. 4, 521–528; MR1928872], Math. Res. Lett. 13 (2006), no. 2-3, 463–465. MR 2231131, DOI 10.4310/MRL.2006.v13.n3.a10
- Charles P. Boyer, Krzysztof Galicki, and Michael Nakamaye, On positive Sasakian geometry, Geom. Dedicata 101 (2003), 93–102. MR 2017897, DOI 10.1023/A:1026363529906
- Diarmuid Crowley and David J. Wraith, Positive Ricci curvature on highly connected manifolds, J. Differential Geom. 106 (2017), no. 2, 187–243. MR 3662991, DOI 10.4310/jdg/1497405625
- Anand Dessai, Some geometric properties of the Witten genus, Alpine perspectives on algebraic topology, Contemp. Math., vol. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 99–115. MR 2581907, DOI 10.1090/conm/504/09877
- Jason DeVito, The classification of compact simply connected biquotients in dimensions 4 and 5, Differential Geom. Appl. 34 (2014), 128–138. MR 3209541, DOI 10.1016/j.difgeo.2014.04.002
- Jason DeVito, The classification of compact simply connected biquotients in dimension 6 and 7, Math. Ann. 368 (2017), no. 3-4, 1493–1541. MR 3673662, DOI 10.1007/s00208-016-1460-8
- Haibao Duan and Chao Liang, Circle bundles over 4-manifolds, Arch. Math. (Basel) 85 (2005), no. 3, 278–282. MR 2172386, DOI 10.1007/s00013-005-1214-4
- Fernando Galaz-Garcia and Martin Kerin, Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension, Math. Z. 276 (2014), no. 1-2, 133–152. MR 3150196, DOI 10.1007/s00209-013-1190-5
- Fernando Galaz-Garcia and Catherine Searle, Low-dimensional manifolds with non-negative curvature and maximal symmetry rank, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2559–2564. MR 2784821, DOI 10.1090/S0002-9939-2010-10655-X
- Peter B. Gilkey, JeongHyeong Park, and Wilderich Tuschmann, Invariant metrics of positive Ricci curvature on principal bundles, Math. Z. 227 (1998), no. 3, 455–463. MR 1612669, DOI 10.1007/PL00004385
- Karsten Grove, Geometry of, and via, symmetries, Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000) Univ. Lecture Ser., vol. 27, Amer. Math. Soc., Providence, RI, 2002, pp. 31–53. MR 1922721, DOI 10.1090/ulect/027/02
- Karsten Grove and Catherine Searle, Positively curved manifolds with maximal symmetry-rank, J. Pure Appl. Algebra 91 (1994), no. 1-3, 137–142. MR 1255926, DOI 10.1016/0022-4049(94)90138-4
- Karsten Grove and Wolfgang Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. Math. 149 (2002), no. 3, 619–646. MR 1923478, DOI 10.1007/s002220200225
- Karsten Grove and Wolfgang Ziller, Polar manifolds and actions, J. Fixed Point Theory Appl. 11 (2012), no. 2, 279–313. MR 3000673, DOI 10.1007/s11784-012-0087-y
- Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
- Akio Hattori and Tomoyoshi Yoshida, Lifting compact group actions in fiber bundles, Japan. J. Math. (N.S.) 2 (1976), no. 1, 13–25. MR 461538, DOI 10.4099/math1924.2.13
- Soon Kyu Kim, Dennis McGavran, and Jingyal Pak, Torus group actions on simply connected manifolds, Pacific J. Math. 53 (1974), 435–444. MR 368051, DOI 10.2140/pjm.1974.53.435
- Shoshichi Kobayashi, Transformation groups in differential geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. MR 1336823
- John C. Nash, Positive Ricci curvature on fibre bundles, J. Differential Geometry 14 (1979), no. 2, 241–254. MR 587552
- Hae Soo Oh, $6$-dimensional manifolds with effective $T^{4}$-actions, Topology Appl. 13 (1982), no. 2, 137–154. MR 644109, DOI 10.1016/0166-8641(82)90016-5
- Hae Soo Oh, Toral actions on $5$-manifolds, Trans. Amer. Math. Soc. 278 (1983), no. 1, 233–252. MR 697072, DOI 10.1090/S0002-9947-1983-0697072-0
- Peter Orlik and Frank Raymond, Actions of $\textrm {SO}(2)$ on 3-manifolds, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 297–318. MR 0263112
- Peter Orlik and Frank Raymond, Actions of the torus on $4$-manifolds. I, Trans. Amer. Math. Soc. 152 (1970), 531–559. MR 268911, DOI 10.1090/S0002-9947-1970-0268911-3
- Frank Raymond, Classification of the actions of the circle on $3$-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51–78. MR 219086, DOI 10.1090/S0002-9947-1968-0219086-9
- Catherine Searle and Frederick Wilhelm, How to lift positive Ricci curvature, Geom. Topol. 19 (2015), no. 3, 1409–1475. MR 3352240, DOI 10.2140/gt.2015.19.1409
- Ji-Ping Sha and DaGang Yang, Positive Ricci curvature on compact simply connected $4$-manifolds, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 529–538. MR 1216644
- Ji-Ping Sha and DaGang Yang, Positive Ricci curvature on the connected sums of $S^n\times S^m$, J. Differential Geom. 33 (1991), no. 1, 127–137. MR 1085137
- Lorenz J. Schwachhöfer and Wilderich Tuschmann, Metrics of positive Ricci curvature on quotient spaces, Math. Ann. 330 (2004), no. 1, 59–91. MR 2091679, DOI 10.1007/s00208-004-0538-x
- Stephan Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Math. Ann. 304 (1996), no. 4, 785–800. MR 1380455, DOI 10.1007/BF01446319
- J. C. Su, Transformation groups on cohomology projective spaces, Trans. Amer. Math. Soc. 106 (1963), 305–318. MR 143839, DOI 10.1090/S0002-9947-1963-0143839-4
- Guofang Wei, Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 311–313. MR 940494, DOI 10.1090/S0273-0979-1988-15653-4
- David Wraith, Surgery on Ricci positive manifolds, J. Reine Angew. Math. 501 (1998), 99–113. MR 1637825, DOI 10.1515/crll.1998.082
- David J. Wraith, New connected sums with positive Ricci curvature, Ann. Global Anal. Geom. 32 (2007), no. 4, 343–360. MR 2346222, DOI 10.1007/s10455-007-9066-8
- David J. Wraith, $G$-manifolds with positive Ricci curvature and many isolated singular orbits, Ann. Global Anal. Geom. 45 (2014), no. 4, 319–335. MR 3180952, DOI 10.1007/s10455-013-9404-y
Additional Information
- Diego Corro
- Affiliation: Institut für Algebra und Geometrie, Karlsruher Institut für Technologie (KIT), 76131 Karlsruhe, Germany
- MR Author ID: 1379059
- ORCID: 0000-0002-1114-0071
- Email: diego.corro@partner.kit.edu
- Fernando Galaz-García
- Affiliation: Institut für Algebra und Geometrie, Karlsruher Institut für Technologie (KIT), 76131 Karlsruhe, Germany
- Address at time of publication: Department of Mathematical Sciences, Durham University, Durham, United Kingdom
- MR Author ID: 822221
- Email: galazgarcia@kit.edu; fernando.galaz-garcia@durham.ac.uk
- Received by editor(s): September 19, 2019
- Received by editor(s) in revised form: December 4, 2019
- Published electronically: March 31, 2020
- Additional Notes: The first author was supported by CONACYT-DAAD (scholarship number 409912).
The second author was supported by the DFG (grant GA 2050 2-1, SPP2026 “Geometry at Infinity”).
Both authors were supported by the DFG (281869850, RTG 2229 “Asymptotic Invariants and Limits of Groups and Spaces”). - Communicated by: Guofang Wei
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3087-3097
- MSC (2010): Primary 53C20, 57S15
- DOI: https://doi.org/10.1090/proc/14961
- MathSciNet review: 4099795