On disk-homogeneous symmetric spaces.

*(English)*Zbl 0533.53047Let (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.).

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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\textit{O. Kowalski} and \textit{L. Vanhecke}, Ann. Global Anal. Geom. 1, No. 2, 91--104 (1983; Zbl 0533.53047)

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##### References:

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[2] | CARPENTER, P., GRAY, A. and WILLMORE, T.J. : The curvature of Einstein symmetric spaces,Quart. J. Math. Oxford, to appear. · Zbl 0509.53045 |

[3] | CHEN, B.Y. and VANHECKE, L.: Differential geometry of geodesic spheres,J. Reine Angew. Math. 325 (1981), 28–67. · Zbl 0503.53013 |

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[5] | KOWALSKI, 0. and VANHECKE, L. : Ball-homogeneous and disk-homogeneous Riemannian manifolds, to appear. · Zbl 0476.53023 |

[6] | RUSE, H.S., WALKER, A.G. and WILLMORE, T.J.:Harmonic spaces, Cremonese, Roma, 1961. · Zbl 0134.39202 |

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