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On disk-homogeneous symmetric spaces. (English) Zbl 0533.53047

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.).
Reviewer: I.Dotti Miatello

MSC:

53C35 Differential geometry of symmetric spaces
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