Let M be a Hadamard manifold, that is, a complete simply connected riemannian manifold with non-positive sectional curvatures. Then every geodesic segment α : [0, a] → M from α(0) to α(a) can be extended to a geodesic ray α : [0,∞) → M . We say then that the Hadamard manifold M is geodesically complete. Note that, in this case, all geodesic rays are proper maps. CAT(0) spaces are generalization...

Based on the ideas of Bessa-Jorge-Montenegro [4] we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as subma...

Suppose that IH is a two-dimensional Cartan-Hadamard manifold with sectional curvatures satisfying a weak negative upper bound and no lower bound. Then the angular part of Brownian motion on IH tends to a limit as time tends to the explosion time of the Brownian motion. Moreover this limit angle has a distribution whose closed support is dense on the circle at infinity. KEY-WORDS : Brownian mot...

Geometry of Fisher metric and geodesics on a space of probability measures defined on a compact manifold is discussed and is applied to geometry of a barycenter map associated with Busemann function on an Hadamard manifold X . We obtain an explicit formula of geodesic and then several theorems on geodesics, one of which asserts that any two probability measures can be joined by a unique geodesi...

The product of two free scalar fields on a manifold is shown to be a well defined operator valued distribution on the GNS Hilbert space of a globally Hadamard product state. Viewed as a new field all n-point distributions exist, giving a new example for a Wightman field on a manifold.

In this paper we consider the proximal point algorithm to approximate a singularity of a multivalued monotone vector field on a Hadamard manifold. We study the convergence of the sequence generated by an inexact form of the algorithm. Our results extend the results of [3, 25] to Hadamard manifolds as well as the main result of [11] with more general assumptions on the control sequence. We also ...

Generalizing [BrFa] we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m1 +m2 −1 in the product X := Xm1 1 ×X m2 2 of two Hadamard manifolds Xmi i of dimension mi with pinched negative curvature. Combining this result with [BuySch] we prove the additivity of the hyperbolic rank for products of manifolds with pinched negative curvature.