Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-24T12:55:39.916Z Has data issue: false hasContentIssue false

Characterizations of Model Manifolds by Means of Certain Differential Systems

Published online by Cambridge University Press:  20 November 2018

S. Pigola
Dipartimento di Fisica e Matematica, Universitá dell’Insubria, 22100 Como, Italy
M. Rimoldi
Dipartimento di Matematica, Universitá degli Studi di Milano, 20133 Milano, Italye-mail: stefano.pigola@uninsubria.ite-mail:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove metric rigidity for complete manifolds supporting solutions of certain second order differential systems, thus extending classical works on a characterization of space-forms. Along the way, we also discover new characterizations of space-forms. We next generalize results concerning metric rigidity via equations involving vector fields.

Research Article
Copyright © Canadian Mathematical Society 2012


[1] Bishop, R. L., Decomposition of cut loci. Proc. Amer. Math. Soc. 65(1977), no. 1, 133136. Google Scholar
[2] Erkekoğlu, F., García-Río, E., Kupeli, D., and Ünal, B., Characterizing specific Riemannian manifolds by differential equations. Acta Appl. Math. 76(2003), no. 2, 195219. Google Scholar
[3] García-Río, E., Kupeli, D., and Ünal, B., On a differential equation characterizing Euclidean spheres. J. Differential Equations 194(2003), no. 2, 287299. Google Scholar
[4] Kanai, M., On a differential equation characterizing a Riemannian structure of a manifold. Tokyo J. Math. 6(1983), no. 1, 143151. Google Scholar
[5] Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14(1962), 333340. Google Scholar
[6] Petersen, P., Riemannian Geometry. Graduate Texts in Mathematics 71. Springer-Verlag, New York, 1998.Google Scholar
[7] Pigola, S., Rigoli, M., and Setti, A. G., Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique. Progress in Mathematics 266. Birkhäuser Verlag, Basel, 2008.Google Scholar
[8] Tashiro, Y., Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117(1965), 251275. Google Scholar
[9] Wolter, F.-E., Distance function and cut loci on a complete Riemannian manifold. Arch. Math. (Basel) 32(1979), no. 1, 9296.Google Scholar