Abstract
We classify complete curvature homogeneous metrics on simply connected four-dimensional manifolds which are invariant under a cohomogeneity one action.
Similar content being viewed by others
References
Alekseevsky, A.V., Alekseevsky, D.V.: \(G\)- manifolds with one dimensional orbit space. Ad. Sov. Math. 8, 1–31 (1992)
Alexandrino, M., Bettiol, R.: Lie Groups and Geometric Aspects of Isometric Actions. Springer, Cham (2015)
Besse, A.: Einstein Manifolds, Modern Surveys in Math, vol. 10. Springer, New York (1987)
Suresh, A., Boeckx, Kowalski, O., Vanhecke L: Riemannian Manifolds of Conullity Two, p. xviii+300xviii+300. World Scientific Publishing Co., River Edge (1996)
Brooks, T.: Three dimensional manifolds with constant Ricci eigenvalues \((\lambda ,\lambda ,0)\), in preparation
Ferus, D., Karcher, H., Münzner, H.F.: Clifford algebras and new isoparametric hypersurfaces. Math. Z. 177, 479–502 (1981)
Grove, K., Ziller, W.: Lifting group actions and nonnegative curvature. Trans. Am. Math. Soc. 363, 2865–2890 (2011)
Grove, K., Verdiani, L., Ziller, W.: An exotic \(T_1{\mathbb{S}}^4\) with positive curvature. Geom. Funct. Anal. 21, 499–524 (2011)
Lax, P.: Linear Algebra and Its Applications, 2nd edn. Wiley-Interscience, New York (2007)
Parker, J.: \(4\)-dimensional \({G}\)-manifolds with \(3\)-dimensional orbits. Pac. J. Math. 125, 187–204 (1986)
Sekigawa, K.: On the Riemannian manifolds of the form \(B_f \times F^n\), Kodai Math. Sem. Rep. 26, 343–347 (1974/75)
Takagi, H.: On curvature homogeneity of Riemannian manifolds. Tohoku Math. J. 26, 581–585 (1974)
Singer, I.M.: Infnitesimally homogeneous spaces. Commun. Pure Appl. Math. 13, 685–697 (1960)
Tsukada, K.: Curvature homogeneous hypersurfaces immersed in a real space form. Tohoku Math. J. 40, 221–244 (1988)
Tricerri, F., Vanhecke, L.: Curvature homogeneous Riemannian manifolds. Ann. Sci. Ec. Norm. Sup. 22, 535–554 (1989)
Kowalski, O., Tricerri, F., Vanhecke, L.: Curvature homogeneous Riemannian manifolds. J. Math. Pures Appl. 71, 471–501 (1992)
Verdiani, L.: Curvature homogeneous metrics of cohomogeneity one. Riv. Mat. Univ. Parma 6, 179–200 (1997)
Verdiani, L., Ziller, W.: Smoothness conditions in cohomogeneity one manifolds. Transf. Groups (2020). https://doi.org/10.1007/s00031-020-09618-9
Ziller, W.: On the geometry of cohomogeneity one manifolds with positive curvature, In: Riemannian Topology and Geometric Structures on Manifolds, in honor of Charles P.Boyer’s 65th birthday, Progress in Mathematics 271, 233–262 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first named author was supported by Prin and GNSAGA grants. The second named author was supported by a grant from the National Science Foundation and by a fellowship from CNPq to support his visit at IMPA.
Rights and permissions
About this article
Cite this article
Verdiani, L., Ziller, W. Curvature Homogeneous Manifolds in Dimension 4. J Geom Anal 31, 8036–8062 (2021). https://doi.org/10.1007/s12220-020-00566-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-020-00566-0