Patrick Eberlein

Isometry groups of simply connected manifolds of nonpositive curvature II

Let H denote a complete simply connected Riemannian manifold of nonpositive sectional curvature, and let I(H) denote the group of isometries of H. In this paper we consider density properties of subgroups D~_I(H) that satisfy the duality condition (defined below), These density properties also yield characterizations of Riemannian symmetric spaces of noncompact type and results about lattices in H that strengthen several of the results of [ 11 ] and [ 15]. If H is a symmetric space of noncompact type and if D is a subgroup of Io(H), then the duality condition for D is implied by the Selberg property (S) for D [20, pp. 4-6] or [10]. A partial converse is obtained in [10]. It is an interesting question whether the two conditions are equivalent in this context. Our density results are very similar to those of [5]. In Proposition 4.2 we obtain a differential geometric version of the Borel density theorem (cf. Corollary 4.2 of [5]): Let H admit no Euclidean de Rham factor, and let G~_I(H) be a subgroup whose normalizer D in I(H) satisfies the duality condition. Then either (1) G is discrete or (2) there exist manifolds Hi , / /2 such that (a) H is isometric to the Riemannian product HlXH2, (b) H1 is a symmetric space of noncompact type, (c) ((~)0=Io(Hl) and (d) there exists a discrete subgroup B~_I(Hz), whose normalizer in 1(//2) satisfies the duality condition, such that Io(HO• is a subgroup of t) of finite index in 0 . Using the result just quoted or the main theorem of section 3 we then obtain the following decomposition of a manifold H whose isometry group I(H) satisfies the duality condition (Proposition 4.1): Let I(H) satisfy the duality condition. Then there exist manifolds H0, Ht and H2, two of which may have dimension zero, such that (1) H is isometric to

Geometry of 2-step nilpotent groups with a left invariant metric. II

We obtain a partial description of the totally geodesic submanifolds of a 2-step, simply connected nilpotent Lie group with a left invariant metric. We consider only the case that N is nonsingular; that is, ad : X Y is surjective for all elements 4 E X Y, where X denotes the Lie algebra of N and Y denotes the center of X. Among other results we show that if H is a totally geodesic submanifold of N with dimH > 1 + dimY, then H is an open subset of gN* , where g is an element of H and N* is a totally geodesic subgroup of N. We find simple and useful criteria that are necessary and sufficient for a subalgebra X* of X to be the Lie algebra of a totally geodesic subgroup N* . We define and study the properties of a Gauss map of a totally geodesic submanifold H of N. We conclude with a characterization of 2-step nilpotent Lie groups N of Heisenberg type in terms of the abundance of totally geodesic submanifolds of N.

A differential geometric characterization of symmetric spaces of higher rank

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Rigidity of lattices of non-positive curvature

Let M, M* denote compact, connected manifolds of non-positive sectional curvature whose fundamental groups are isomorphic and whose Euclidean de Rham factors are trivial. We prove that: if M is a compact irreducible quotient of a reducible symmetric space H, then M and M* are isometric up to a constant multiple of the metric; and that the number and dimensions of the local de Rham factors are the same for M and M*. Gromov has independently proved the first result in the more general case that M is locally symmetric and globally irreducible with rank at least two. 0. Introduction A basic problem in Riemannian geometry is to determine the extent to which the geometry of the Riemannian metric and the topology of the underlying manifold influence each other. Restrictions on the curvature and topology are usually necessary to obtain reasonable results, and we shall confine our attention to compact connected manifolds of nonpositive sectional curvature. We define a geometric property of such manifolds to be a rigid property if whenever it holds for a manifold M it also holds for any manifold M* that is homotopically equivalent to M. Our goal is to look for rigid properties. In a previous paper [7] we showed that certain geometric properties of a free homotopy class of closed curves are rigid properties. In this paper we have two main results, the first of which is half of an independent result of Gromov. Before stating them we define a Riemannian manifold X to be reducible if some finite Riemannian cover X splits as a nontrivial Riemannian product X\ x X2. If X is simply connected and reducible, then X itself is a nontrivial Riemannian product THEOREM A. Let M, M* denote compact connected Riemannian manifolds of nonpositive sectional curvature whose fundamental groups are isomorphic and whose universal Riemannian covering manifolds H, H* possess no Euclidean de Rham factor. Suppose that H* is a reducible symmetric space of noncompact type and M* is an irreducible quotient of H*. Then M and M* are isometric, provided that one multiplies the metric of M or M* by a suitable positive constant.

Conjugacy and rigidity for nonpositively curved manifolds of higher rank

Abstract Let M and N be compact Riemannian manifolds with sectional curvature K ⩽ 0 such that M has dimension ⩾ 3 and rank ⩾ 2. If there exists a C 0 conjugacy F between the geodesic flows of the unit tangent bundles of M and N , then there exists an isometry G : M → N that induces the same isomorphism as F between the fundamental groups of M and N .

Symmetry diffeomorphism group of a manifold of nonpositive curvature

Let M denote a complete simply connected manifold of nonpos- itive sectional curvature. For each point p S M let sp denote the diffeomor- phism of M that fixes p and reverses all geodesies through p. The symmetry diffeomorphism group G* generated by all diffeomorphisms (sp: p € M} ex- tends naturally to group of homeomorphisms of the boundary sphere M(oo). A subset X of M(oo) is called involutive if it is invariant under G*. THEOREM. Let X C M(co) be a proper, closed involutive subset. For each point p S M let N(p) denote the linear span in TPM of those vectors at p that are tangent to a geodesic 7 whose asymptotic equivalence class 7(00) belongs to X. If N(p) is a proper subspace of TPM for some point p € M, then M splits as a Riemannian product My x Mi such that N is the distribution of M induced by My. This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if M is irreducible and M(oo) admits a proper, closed involutive subset X, then M is isometric to a symmetric space of noncompact type and rank k>2.