On the geometric rank of homogeneous spaces of nonpositive curvature
Let \(H\) denote a complete simply connected Riemannian manifold of sectional curvature \(K \leq 0\). \(H\) is always diffeomorphic to a Euclidean space of the same dimension. The geometric rank \(r\) of \(H\) is the largest integer \(r\) for which every geodesic of \(H\) is contained in a complete, totally geodesic imbedded copy of \(\mathbb{R}^ r\). If \(H\) is irreducible of geometric rank \(r \geq 2\), has a lower bound for the sectional curvature and admits a quotient manifold of finite volume, then W. Ballmann and K. Burns, R. Spatzier proved independently that \(H\) must be Riemannian symmetric. The author and reviewer of this article later generalized this result and in particular removed the hypothesis of a lower bound for the sectional curvature. If \(H\) is a homogeneous space, then \(H\) admits a simply transitive solvable group \(S\) of isometries, and \(H\) is isometric to \(S\) with an appropriate left invariant metric. The group \(S\) is not uniquely determined, but the codimension \(k\) of the commutator subgroup \([S,S]\) as a submanifold of \(S\) is independent of the choice of \(S\). The integer \(k\) is the algebraic rank of \(H\) and is always at least as large as the geometric rank \(r\). The main result of this article is the following analogue for homogeneous spaces of the Rank Rigidity Theorem of Ballman and Burns-Spatzier. Theorem. Let \(H\) as above be homogeneous and irreducible with geometric rank \(r \geq 2\). Then \(H\) is a Riemannian symmetric space of noncompact type. Combining this result with a result of J. Wolf one obtains the following Corollary. Every homogeneous space of sectional curvature \(K \leq 0\) admits a unique decomposition (up to order) as a Riemannian product \(T \times E \times S \times H_ 1 \times \dots \times H_ m\), where \(T\) is a flat torus, \(E\) is a Euclidean space, \(S\) is a symmetric space of noncompact type and the factors \(H_ i\) are irreducible and homogeneous with geometric rank 1. Any of the factors may be absent. There is an important essential difference between this article and previous ones; a homogeneous space \(H\) never admits a quotient manifold of finite volume unless \(H\) is Riemannian symmetric. We describe some of the main ideas. To every manifold \(H\) (not necessarily homogeneous) one may attach a boundary sphere \(H(\infty)\) consisting of equivalence classes of asymptotic geodesics of \(H\). The space \(H(\infty)\) admits a natural Tits pseudometric Td introduced by Gromov, and Td may take on the value \(+\infty\). If \(H\) is Riemannian symmetric with geometric rank \(r \geq 2\), then \(\text{Td}(x,y) \leq \pi\) for all \(x,y \in H(\infty)\) with equality for the endpoints in \(H(\infty)\) of any geodesic of \(H\). In this case one may define a Tits building structure in the metric space \((H(\infty),\text{Td})\) that carries the same information as the standard Tits building structure for the semisimple group \(G = I_ 0(H)\), the connected isometry group of \(H\). One of the main steps in this article is to generalize the Tits building structure of \(H(\infty)\) for symmetric spaces \(H\) to a \((k-1)\)-dimensional spherical polyhedral structure of \(H(\infty)\) for homogeneous spaces \(H\) with algebraic rank \(k\). Under the hypotheses of the main theorem the author proves the existence of singular geodesics in \(H\) and then modifies ideas of Ballmann to deduce the main theorem from the Holonomy Theorem of Berger. The author also shows that the finiteness of the metric Td is not enough to characterize the irreducible symmetric spaces among the irreducible homogeneous spaces. For each \(\varepsilon > 0\) there exists a homogeneous nonsymmetric space \(H\) with geometric rank \(r = 1\) and algebraic rank \(k \geq 2\) such that \(\text{Td}(x,y) \leq \pi + \varepsilon\) for all \(x,y \in H(\infty)\).
scientific article; zbMATH DE number 447096
scientific article