The De Rham Decomposition Theorem For Metric Spaces

Abstract

We generalize the classical de Rham decomposition theorem for Riemannian manifolds to the setting of geodesic metric spaces of finite dimension.

At infinity of finite-dimensional CAT(0) spaces

“…It was shown by FoertschLytchak [11] that any finite-dimensional CAT(0) space (and more generally any geodesic metric space of finite affine rank) admits a canonical isometric splitting into a flat factor and finitely many non-flat irreducible factors. Building upon [11], it was then shown by Caprace-Monod [7,Corollary 4.3(ii)] that the same conclusion holds for proper CAT(0) spaces whose isometry group acts minimally, assuming that the Tits boundary is finite-dimensional. We shall need the following 'improper' variation of this result.…”

Some properties of non-positively curved lattices

“…Next, we establish a group decomposition, supplemented by a de Rham decomposition of the space which is a variant of [4]: q (p, q, n 0) where S i are almost connected simple Lie groups with trivial centre and D j are totally disconnected irreducible groups. Furthermore, there is a canonical equivariant isometric splitting…”

“…On se place dans le cadre où un groupe G < Is(X) agit minimalement et sans point fixe à l'infini (il convient de montrer qu'il est possible de se restreindre à ce cas). Lorsque le bord à l'infini de X, muni de la métrique de Tits, est de dimension finie, on montre que X possède une décomposition canonique en un produit d'un facteur euclidien et d'un nombre fini de facteurs irréductibles non euclidiens ; c'est là une variante de la décomposition de de Rham obtenue dans [4]. En outre le groupe d'isométries complet de X se décompose virtuellement comme produit des groupes d'isométries de chaque facteur de X.…”

Large scale detection of half-flats in CAT(0)-spaces

“…Furthermore, this decomposition is unique up to permutation of the factors. This result was recently generalized by Foertsch-Lytchak [13] to cover finite dimensional geodesic metric spaces (such as ultralimits of Riemannian manifolds). Our next corollary shows that, in the presence of non-positive Riemannian curvature, there is a strong relationship between splittings ofM and splittings of Cone(M).…”