Hilbert space factor of metric spaces
We prove that any complete metric space has a unique decomposition as a direct product of a possibly finite or zero-dimensional Hilbert space and a space that does not split off lines.
math.DG↗
arXiv subjects
Thomas Foertsch
Publications and source records attributed to Thomas Foertsch.
A Möbius Characterization of Metric Spheres
In this paper we characterize compact extended Ptolemy metric spaces with many circles up to Möbius equivalence. This characterization yields a Möbius characterization of the $n$-dimensional spheres $S^n$ and hemispheres $S^n_+$ when endowed with their chordal metrics. In particular, we show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to $(S^n,d_0)$ for some $n\ge 1$, the $n$-dimensional sphere $S^n$ with its chordal metric.
math.MG↗
Group actions on geodesic Ptolemy spaces
In this paper we study geodesic Ptolemy metric spaces $X$ which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that $X$ is equivariantly rough isometric to a Euclidean space.
math.MG↗
Non-positive curvature and the Ptolemy inequality
We provide examples of non-locally compact geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.
math.MG↗
Hyperbolicity, CAT(-1)-spaces and the Ptolemy Inequality
Using a four points inequality for the boundary of CAT(-1)-spaces, we study the relation between Gromov hyperbolic spaces and CAT(-1)-spaces.
math.MG↗
The de Rham decomposition theorem for metric spaces
We generalize the classical de Rham decomposition theorem for Riemannian manifolds to the setting of geodesic metric spaces of finite dimension.
math.MG↗
Hilbert metrics and Minkowski norms
It is shown that the Hilbert geometry $(D,h_D)$ associated to a bounded convex domain $D\subset \mathbb{E}^n$ is isometric to a normed vector space $(V,||\cdot ||)$ if and only if $D$ is an open $n$-simplex. One further result on the asymptotic geometry of Hilbert's metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.
math.MG↗
Isometries of spaces of convex compact subsets of CAT(0)-spaces
In the present paper we characterize the surjective isometries of the space of compact, convex subsets of proper, geodesically complete CAT(0)-spaces in which geodesics do not split, endowed with the Hausdorff metric. Moreover, an analogue characterization of the surjective isometries of the space of compact subsets of a proper, uniquely geodesic, geodesically complete metric space in which geodesics do not split, when endowed with the Hausdorff metric, is given.
math.MG↗
Products of hyperbolic metric spaces II
In arXiv math.MG/0207296 we introduced a product construction for locally compact, complete, geodesic hyperbolic metric spaces. In the present paper we define the hyperbolic product for general Gromov-hyperbolic spaces. In the case of roughly geodesic spaces we also analyse the boundary at infinity.
math.MG↗
Inertial forces and photon surfaces in arbitrary spacetimes
Given, in an arbitrary spacetime, a 2-dimensional timelike submanifold (worldsheet) and an observer field on this worldsheet, we assign gravitational, centrifugal, Coriolis and Euler forces to every particle worldline in the worldsheet with respect to the observer field. We prove that centrifugal and Coriolis forces vanish, for all particle worldlines with respect to any observer field, if and only if the worldsheet is a photon 2-surface, i.e., generated by two families of lightlike geodesics. We further demonstrate that a photon 2-surface can be characterized in terms of gyroscope transport and we give several mathematical criteria for the existence of photon 2-surfaces. Finally, examples of photon 2-surfaces in conformally flat spacetimes, in Schwarzschild and Reissner-Nordstroem spacetimes, and in Goedel spacetime are worked out.
gr-qc↗
Non Standard Metric Products
We consider a fairly general class of natural non standard metric products and classify those amongst them, which yield a product of certain type (for instance an inner metric space) for all possible choices of factors of this type (inner metric spaces). We further prove the additivity of the Minkowski rank for a large class of metric products.
math.MG↗
Hyperbolic Rank of Products
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.
math.DG↗
Products of hyperbolic metric spaces
Let (X_i,d_i), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a ``hyperbolic product'' X_1{times}_h X_2 which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.
math.MG↗
Minkowski- versus Euclidean rank for products of metric spaces
We introduce a notion of the Euclidean- and the Minkowski rank for arbitrary metric spaces and we study their behaviour with respect to products. We show that the Minkowski rank is additive with respect to metric products, while additivity of the Euclidean rank only holds under additional assumptions, e.g. for Riemannian manifolds. We also study products with nonstandard product metrics.
math.MG↗