Abstract
We present an efficient method for determining the conditions that a metric on a cohomogeneity one manifold, defined in terms of functions on the regular part, needs to satisfy in order to extend smoothly to the singular orbit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. V. Alekseevsky, D. V. Alekseevsky, G-manifolds with one-dimensional orbit space, Adv. in Sov. Math. 8 (1992), 1–31.
M. Alexandrino, R. Bettiol, Lie Groups and Geometric Aspects of Isometric Actions, Springer, Cham, 2015.
H. Chi, Invariant Ricci at metrics of cohomogeneity one with Wallach spaces as principal orbits, arXive:1903.01641v1 (2019).
A. Dancer, M. Wang, On Ricci solitons of cohomogeneity one, Ann. Glob. Anal. Geom. 39 (2011), 259–292.
J. H. Eschenburg, M. Wang, The initial value problem for cohomogeneity one Einstein metrics, J. Geom. Anal. 10 (2000), 109–137.
L. Foscolo, M. Haskins, New G2-holonomy cones and exotic nearly Khler structures on S6 and S3 × S3 × S3, Ann. of Math. 185 (2017), 59–130.
S. Goette, M. Kerin, K. Shankar, Highly connected 7-manifolds and non-negative sectional curvature, Ann. of Math., to appear.
K. Grove, L. Verdiani, W. Ziller, An exotic T1S4 with positive curvature, Geom. Funct. Anal. 21 (2011), 499–524.
K. Grove, L. Verdiani, B. Wilking, W. Ziller, Non-negative curvature obstruction in cohomogeneity one and the Kervaire spheres, Ann. del. Scuola Norm. Sup. 5 (2006), 159–170.
K. Grove, W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Inv. Math. 149 (2002), 619–646.
J. Kazdan, F. Warner, Curvature functions for open 2-manifolds Ann. of Math. 99 (1974), 203–219.
N. Koiso, Y. Sakane, Nonhomogeneous Kähler–Einstein metrics on compact complex manifolds, in: Curvature and Topology of Riemannian Manifolds (Katata, 1985), Lecture Notes in Math., Vol. 1201, Springer, Berlin, 1986, pp. 165–179; Part II, Osaka J. Math. 25 (1988), 933–959.
L. Verdiani, Invariant metrics on cohomogeneity one manifolds, Geom. Dedicata 77 (1999), 77–111.
L. Verdiani, W. Ziller, Concavity and rigidity in non-negative curvature, J. Diff. Geom. 97 (2014), 349–375.
L. Verdiani, W. Ziller, Seven-dimensional cohomogeneity one manifolds with nonnegative curvature, Math. Ann 371 (2018), 655–652.
L. Verdiani, W. Ziller, Four-dimensional curvature homogeneous cohomogeneity one metrics, in preparation.
L. Verdiani, W. Ziller, On the initial value problem for Einstein metrics on cohomogeneity one manifolds, in preparation.
W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann. 259 (1982), 351–358.
Author information
Authors and Affiliations
Corresponding author
Additional information
Funding Information
Open access funding provided by Università degli Studi di Firenze within the CRUI-CARE Agreement.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
L. Verdiani is supported by PRIN and GNSAGA grants.
W. Ziller is supported by a grant from the National Science Foundation.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
VERDIANI, L., ZILLER, W. SMOOTHNESS CONDITIONS IN COHOMOGENEITY ONE MANIFOLDS. Transformation Groups 27, 311–342 (2022). https://doi.org/10.1007/s00031-020-09618-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-020-09618-9