<ENGLISH> Consider a closed, smooth manifold M of nonpositive sectional curvature. Write p:UM-> M for the unit tangent bundle over M and let R_> denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow on UM. We will define the structured dimension sdim(R_>) which, essentially, is the dimension of the set p(R_>) of base points of R_>. The main result holds for manifolds with sdim(R_>) < dim(M)/2: For every E>0 there is an E-dense, flow invariant, closed subset Z_E in UM R_> such that p(Z_E)=M. For every point in M this means that through this point there is a complete geodesic for which the velocity vector field avoids a neighbourhood of R_>.
<DEUTSCH> Gegeben sei eine geschlossene, glatte Mannigfaltigkeit M nichtpositiver Schnittkruemmung. Das Einheitstangentialbuendel sei mit p:UM-> M bezeichnet und die Teilmenge aller Vektoren hoeheren Ranges mit R_>. Diese Teilmenge ist abgeschlossen und invariant unter dem geodaetischen Fluss auf UM. Wir definieren die Strukturdimension sdim(R_>) von R_>, die, im Wesentlichen, die Dimension der Fusspunktmenge p(R_>) misst. Das Hauptergebnis gilt unter der Bedingung, dass sdim(R_>) < dim(M)/2 gilt: Fuer jedes E>0 gibt es eine E-dichte, flussinvariante, abgeschlossene Teilmenge Z_E in UM R_>, fuer die gilt p(Z_E) = M. Dies bedeutet, dass es durch jeden Punkt in M eine vollstaendige Geodaete gibt, deren Geschwindigkeitsfeld eine Umgebung von R_> vermeidet.
In a previous paper, we proved that a quasi-isometric map $f:X\longrightarrow Y$ between two pinched Hadamard manifolds $X$ and $Y$ is within bounded distance from a unique harmonic map. We extend this result to maps $f:Γ\backslash X\longrightarrow Y$, where $Γ$ is a convex cocompact discrete group of isometries of $X$ and $f$ is locally quasi-isometric at infinity.
In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to eigenvalue zero is investigated with respect to rank of geodesics. On a harmonic Hadamard manifold which is of purely exponential volume growth, or of hypergeometric type it is shown that every Busemann function admits positive definite Hessian. A criterion for $(M,g)$ fulfilling …
In this paper we study a homological version of the higher-dimensional divergence invariants defined by Brady and Farb. We show that they are quasi-isometry invariants in the class of proper cocompact Hadamard spaces in the sense of Alexandrov and that they can moreover be used to detect the Euclidean rank of such spaces. We thereby extend results of Brady-Farb, Leuzinger, and Hindawi from the setting of symmetric spaces of non-compact type to that of singular Hadamard spaces. Finally, we exhibit the optimal power for the growth of the divergence above the rank for symmetric spaces of non-comp…
It is well known that simply connected symmetric spaces of non-positive sectional curvature admit a linear isoperimetric filling inequality for cycles of dimension greater than or equal to the rank of the space. In this note we extend that result to homogeneous Hadamard manifolds.
We study the filling invariants at infinity div_{k} for Hadamard manifolds defined by Brady and Farb in [Filling-invariants at infinity for manifolds of nonpositive curvature]. Among other results, we give a positive answer to the question they posed: whether these invariants can be used to detect the rank of a symmetric space of noncompact type.
Correlation matrices are used in many domains of neurosciences such as fMRI, EEG, MEG. However, statistical analyses often rely on embeddings into a Euclidean space or into Symmetric Positive Definite matrices which do not provide intrinsic tools. The quotient-affine metric was recently introduced as the quotient of the affine-invariant metric on SPD matrices by the action of diagonal matrices. In this work, we provide most of the fundamental Riemannian operations of the quotient-affine metric: the expression of the metric itself, the geodesics with initial tangent vector, the Levi-Civita conn…
In a Riemannian manifold a regular convex domain is said to be $λ$-convex if its normal curvature at each point is greater than or equal to $λ$. In a Hadamard manifold, the asymptotic behaviour of the quotient $\vol(Ω(t))/\vol(\partialΩ(t))$ for a family of $λ$-convex domains $Ω(t)$ expanding over the whole space has been studied and general bounds for this quotient are known. In this paper we improve this general result in the complex hyperbolic space $\CH$, a Hadamard manifold with constant holomorphic curvature equal to $-4k^2$. Furthermore, we give some specific properties of convex domain…
Generalizing results due to Brady and Farb we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m_1+m_2-1 in the product X:=X_1^{m_1} times X_2^{m_2} of two Hadamard manifolds X_i^{m_i} of dimension m_i with pinched negative curvature. Combining this result with a Theorem by Buyalo and Schroeder we prove the additivity of the hyperbolic rank for products of manifolds with pinched negative curvature.
We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Every such geometrically infinite isometry subgroup $Γ$ has a set of nonconical limit points with cardinality of continuum.
Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answ…
This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is the one induced by the product of the Stiefel manifold and the manifold of Hermitian positive definite matrices, quotiented by the unitary group. One of the main contribution is to consider an original Riemannian metric, leading to new representations of tangent spaces and geodesics. From this geometry, we derive a new Riemannian optimization framework for r…
In contrast to SPD matrices, few tools exist to perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as the quotient of the affine-invariant metric by the congruence action of positive diagonal matrices. The space of SPD matrices had always been thought of as a Riemannian homogeneous space. In contrast, we view in this work SPD matrices as a Lie group and the affine-invariant metric as a left-invariant metric. This unexpected new viewpoint allows us to generalize the construction of the quotient-affine metric and t…
Let $X$ be a Hadamard manifold, and $Γ$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/Γ$ to the behavior of the Poincar{é} series of $Γ$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $Γ$-invariant conformal densities supported on the geometric limit set of $Γ$.
This belongs to a series of papers motivated by Ballmann's Higher Rank Rigidity Conjecture. We prove the following. Let $X$ be a CAT(0) space with a geometric group action. Suppose that every geodesic in $X$ lies in an $n$-flat, $n\geq 2$. If $X$ contains a periodic $n$-flat which does not bound a flat $(n+1)$-half-space, then $X$ is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.
