On disk-homogeneous symmetric spaces
type: Item
instance of: scholarly article
title: On disk-homogeneous symmetric spaces
zbMATH Open document ID: 0533.53047
DOI: 10.1007/BF02330007
published in: Annals of Global Analysis and Geometry
publication date: 1983-01-01T00:00:00Z
review text: Let (M,g) be a connected analytic Riemannian manifold of dimension \(\geq 3\). If \(x\in T_ mM\) is a unit tangent vector let \(V^ x\!_ m(r)\) denote the (n-1) dimensional volume of \(D^ x\!_ m(r)=\{m'\in M| d(m',m)\leq r\}\cap \exp_ m(\{x\}^{\perp})\) where \(r<i(m)\) and i(m) is the injectivity radius at m. The main result proved by the authors in the present paper is Theorem 1. Let M be a connected locally symmetric space of dimension \(\geq 3\). Then M is strongly disk homogeneous up to order 6, i.e. the volume \(V^ x\!_ m(r)\) of a disk satisfies \(V^ x\!_ m(r)=V_ 0\!^{n-1}(r)\{1+ar^ 2+br^ 4+cr^ 6+O(r^ 8)\}\) with constants a, b and c, if and only if one of the following cases occurs: i) M is locally flat; ii) M is locally isometric to a rank one symmetric space; iii) M is locally isometric to the Lie group \(E_ 8\), the symmetric space \(E_ 8/D_ 8\), the symmetric space \(E_{8(-24)}\) or to some of their noncompact duals. The authors furthermore conjecture that if M is strongly disk homogeneous up to order 8 then only i) or ii) above occur. By analyzing a power series expansion up to order 6 for \(V^ x\!_ m(r)\) (obtained by the authors in another article) they obtain. Theorem 2. An analytic Riemannian manifold of dimension \(\geq 3\) is strongly disk homogeneous up to order 4 if and only if it is a 2-Stein space (i.e. trace \(R_ x\) and trace \(R^ 2\!_ x\) are constants). If this is the case then (M,g) is irreducible or locally Euclidean. Proposition 9. For a locally symmetric space (M,g) of dimension \(\geq 3\) the following two conditions are equivalent: a) M is strongly disk homogeneous up to order 6; b) (M,g) is a 3-Stein space (i.e. 2-Stein and trace \(R^ 3\!_ x=cons\tan t)\). Finally theorem 1 follows from proposition 9 above and a result by Carpenter, Gray and Willmore (to appear in Q. J. Math., Oxf., II. Ser.).
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 3846415
zbMATH Keywords: strongly disk homogeneous
zbMATH Keywords: locally flat
zbMATH Keywords: symmetric space
zbMATH Keywords: Lie group
zbMATH Keywords: 2- Stein space
author: Oldřich Kowalski
author: Lieven Vanhecke
reviewed by: Isabel Dotti Miatello
MaRDI profile type: Publication
cites work: Q4169382
cites work: THE CURVATURE OF EINSTEIN SYMMETRIC SPACES
cites work: Differential geometry of geodesic spheres.
cites work: Riemannian geometry as determined by the volumes of small geodesic balls
cites work: Ball-homogeneous and disk-homogeneous Riemannian manifolds
cites work: Q5345507
full work available at URL: https://doi.org/10.1007/bf02330007
OpenAlex ID: W2012709504
description: scientific article; zbMATH DE number 3846415