A contribution to the Cartan's method of specialization of frames
A metric characterization of manifolds with boundary
A new definition of the circle by the use of bisectors
An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature
Ball versus distance convexity of metric spaces.
Busemann Functions and Total Curvature.
Caristi's fixed point theorem and metric convexity
Connections and applications of some tangency relation of sets.
Detecting codimension one manifold factors with the piecewise disjoint arc-disk property and related properties
We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note that in every example presently known to the authors of a codimension one manifold factor of dimension n ≥ 4 determined by general position properties, the piecewise disjoint arc-disk...
Differentiability of Busemann Functions and Total Excess.
Flats in Spaces with Convex Geodesic Bicombings
In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales.We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov’s hyperbolicity criterion referring to embedded planes. This generalizes...
Fundamental domains in Teichmüller space.
Geodesics in Asymmetic Metric Spaces
In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We...
Isometries of spaces of convex compact subsets of globally non-positively Busemann curved spaces
We consider the Hausdorff metric on the space of compact convex subsets of a proper, geodesically complete metric space of globally non-positive Busemann curvature in which geodesics do not split, and characterize their surjective isometries. Moreover, an analogous 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 is given.
Length of curves on Lip manifolds
In this paper the length of a curve on a Lipschitz Riemannian manifold is defined. It is shown that the above definition is consistent with the definition of the geodesic distance already introduced by the authors, both in a geometrical and analytical way.
Nonsymmetric weakly complete G-spaces
O čtvrtém Hilbertově problému
On Asymmetric Distances
In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the...
On conditions under which local isometries are motions
On generalizations of pseudo-harmonic and pseudo-Killing vectors