On cohomogeneity one flat Riemannian manifolds

R. Mirzaie; S.M.B. Kashani

doi:10.1017/S0017089502020189

Abstract

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 study the topological properties of cohomogeneity one flat manifolds and their orbits. Among other results we prove that principal orbits of R^n are isometric to R^{n-1} or S^k(c)\times R^{n-k-1}. We show that if M has one singular orbit, it is a totally geodesic submanifold of M and if M is orientable then there is at most one singular orbit.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Mirzaie, R. 2007. Cohomogeneity Two Actions on Flat Riemannian Manifolds. Acta Mathematica Sinica, English Series, Vol. 23, Issue. 9, p. 1587.

Ahmadi, Parviz 2010. On groups acting by cohomogeneity one on the euclidean space ℝ n. Journal of Contemporary Mathematical Analysis, Vol. 45, Issue. 3, p. 174.

MIRZAIE, Reza 2011. ON ORBITS OF ISOMETRIC ACTIONS ON FLAT RIEMANNIAN MANIFOLDS. Kyushu Journal of Mathematics, Vol. 65, Issue. 2, p. 383.

2012. Harmonic Vector Fields. p. 491.

Ahmadi, P. 2015. Singular Orbits in Cohomogeneity One Pseudo-Euclidean Space $$\mathbb {R}^4_\nu $$ R ν 4. Bulletin of the Malaysian Mathematical Sciences Society, Vol. 38, Issue. 1, p. 73.

Mirzaie, R. and Sorush, H. 2016. Topology of orbits and orbit spaces of some product G-manifolds. Manuscripta Mathematica, Vol. 149, Issue. 3-4, p. 297.

Heidari, M. Mirzaie, R. and Srivastava, Hari M. 2018. Universally polar cohomogeneity two Riemannian manifolds of constant negative curvature. Cogent Mathematics & Statistics, Vol. 5, Issue. 1, p. 1523516.

Ahmadi, P. 2019. Cohomogeneity One Dynamics on Three Dimensional Minkowski Space. Zurnal matematiceskoj fiziki, analiza, geometrii, Vol. 15, Issue. 2, p. 155.

Hassani, M. and Ahmadi, P. 2020. Cohomogeneity one actions on the three-dimensional Einstein universe. Geometriae Dedicata, Vol. 206, Issue. 1, p. 105.

Ahmadi, P. and Safari, S. 2020. On cohomogeneity one linear actions on pseudo-Euclidean space Rp,q. Differential Geometry and its Applications, Vol. 68, Issue. , p. 101584.

Mirzaie, Reza and Bakhtiari, Marzie 2021. Cohomogeneity Two Nonsemisimple Isometric Actions. Bulletin of the Iranian Mathematical Society, Vol. 47, Issue. 2, p. 585.

Ahmadi, Parviz Safari, Salim and Hassani, Masoud 2021. A Classification of Cohomogeneity One Actions on the Minkowski Space $$\mathbb {R}^{3,1}$$. Bulletin of the Iranian Mathematical Society, Vol. 47, Issue. 6, p. 1905.

Frenck, Georg Galaz‐García, Fernando and Reiser, Philipp 2022. Cohomogeneity one manifolds and homogeneous spaces of positive scalar curvature. Bulletin of the London Mathematical Society, Vol. 54, Issue. 1, p. 71.

Ahmadi, Parviz and Safari, Salim 2022. Actions on the Four-Dimensional Pseudo-Euclidean Space ℝ2,2 with a Three-Dimensional Orbit. Zurnal matematiceskoj fiziki, analiza, geometrii, Vol. 18, Issue. 3, p. 303.

Hassani, M. and Ahmadi, P. 2023. Isometric actions on the four dimensional Minkowski spacetime. Differential Geometry and its Applications, Vol. 88, Issue. , p. 102007.

Article contents

On cohomogeneity one flat Riemannian manifolds

Abstract

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests