arXiv subjects
Publications and source records attributed to Alexander Lytchak.
We prove that the covering dimension of a metric space with curvature bounded above equals its geometric dimension, without a separability assumption. This resolves a conjecture formulated by Kleiner that goes back to Gromov's work.
We prove that there are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, fixed dimension $n\neq5$, and finite fundamental group, up to foliated diffeomorphism. In addition, we construct various infinite families of isoparametric foliations that are mutually not foliated diffeomorphic, for instance on a fixed sphere.
We prove that any convex flat subset in a complete Euclidean building is contained in an apartment of the maximal system of apartments.
We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases and conjectured in general. In application, we rule out stickiness, while providing examples of nowhere smooth Fréchet functions and we discuss extensions of the statement to Fréchet $p$-means, for $p\neq 2$, as well as to noncomplete manifolds and more general 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.
We prove that a Riemannian submersion between smooth, compact, non-negatively curved Riemannian manifolds has to be smooth, resolving a conjecture by Berestovskii--Guijarro. We show that without any curvature assumption, the smoothness of the base is implied by the smoothness of the total space. Results are proven in the much more general setting of submetries. These are metric generalizations of Riemannian submersions and isometric actions, which have recently appeared in different areas of geometry.
We discuss regularity statements for equidistant decompositions of Riemannian manifolds and for the corresponding quotient spaces. We show that any stratum of the quotient space has curvature locally bounded from both sides.
We show that a length-minimizing disk inherites the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs inherit the upper curvature bound from the ambient space.
We provide new structural results on sets of positive reach in Euclidean spaces and Riemannian manifolds.
We discuss solutions of several questions concerning the geometry of conformal planes.
We analyze weak convergence on CAT(0) spaces and the existence and properties of corresponding weak topologies.
We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.
We prove that a topological 4-manifold of globally non-positive curvature is homeomorphic to Euclidean space.
We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M,g). For example, if a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.
We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall. Our construction uses a tractrix flow which can be defined as a gradient flow for a family of functions of certain type. In an appendix we prove a general existence result for gradient flows of time-dependent locally Lipschitz semiconcave functions, which is of independent interest.
We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.
We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.
We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.