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Pedro Zühlke

Publications and source records attributed to Pedro Zühlke.

8 recordsLinked to original sources
arXiv2025

Homotopy type of spaces of curves with constrained curvature on flat surfaces

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an $n$-sphere, and every $n\geq 1$ is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.

math.GT↗
arXiv2018

On a class of immersions of spheres into space forms of nonpositive curvature

Let $ M^{n+1} $ ($ n \ge 2 $) be a simply-connected space form of sectional curvature $ -κ^2 $ for some $ κ\geq 0 $, and $ I $ an interval not containing $ [-κ,κ] $ in its interior. It is known that the domain of a closed immersed hypersurface of $ M $ whose principal curvatures lie in $ I $ must be diffeomorphic to the sphere $ S^n $. These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed $ I $, the space $ \mathscr{F} $ of all such closed hypersurfaces is either empty or weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of $ S^n $. An equivalence assigns to each element of $ \mathscr{F} $ a suitable modification of its Gauss map. For $ M $ not simply-connected, $ \mathscr{F} $ is the quotient of the corresponding space of hypersurfaces of the universal cover of $ M $ by a natural free proper action of the fundamental group of $ M $.

math.GT↗
arXiv2018

Homotopical and topological rigidity of hypersurfaces of spherical space forms

The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms which extends similar results due to do Carmo/Warner, Wang/Xia and Longa/Ripoll. Under certain sharp conditions on the principal curvatures of such a hypersurface $ f \colon N^n \to M^{n+1} $ $( n\ge 2 )$, it asserts that the universal cover of $ N $ must be diffeomorphic to the $ n $-sphere $ {S}^n $, and provides an upper bound for the order of the fundamental group of $ N $ in terms of that of $ M $. In particular, if $ M = {S}^{n+1} $, then $ N $ is diffeomorphic to $ {S}^n $ and either $ f $ or its Gauss map is an embedding. Let $ J \subset (0,π) $ be any interval of length less than $ \fracπ{2} $. The second main result constructs a weak homotopy equivalence between the space of all complete immersed hypersurfaces of $ M $ with principal curvatures in $ \cot (J) $ and the twisted product of $ \big( Γ\backslash \mathrm{SO}_{n+2} \big) $ and $ \mathrm{Diff}_+({S}^n) $ by $ \mathrm{SO}_{n+1} $, where $ Γ$ is the fundamental group of $ M $ regarded as a subgroup of $ \mathrm{SO}_{n+2} $. Relying on another rigidity criterion due to Wang/Xia, the third main result constructs a homotopy equivalence between the space of all complete immersed hypersurfaces of $ S^{n+1} $ whose Gauss maps have image contained in a strictly convex ball and the same twisted product, with $ Γ$ the trivial group.

math.GT↗
arXiv2018

Spaces of curves with constrained curvature on hyperbolic surfaces

Let $ S $ be a hyperbolic surface. We investigate the topology of the space of all curves on $ S $ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval $ (κ_1,κ_2) $. Such a space falls into one of four qualitatively distinct classes, according to whether $ (κ_1,κ_2) $ contains, overlaps, is disjoint from, or contained in the interval $ [-1,1] $. Its homotopy type is computed in the latter two cases. We also study the behavior of these spaces under covering maps when $ S $ is arbitrary (not necessarily hyperbolic nor orientable) and show that if $ S $ is compact then they are always nonempty.

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arXiv2015

On a discrete version of length metrics

Let $ (X,d) $ be a metric space. We study a metric $ d_0 $ on $ X $ naturally derived from $ d $. If $ (X,d) $ is complete and locally compact, or if it is complete and $ (d_0)_0=d_0 $, then $ d_0 $ coincides with the length metric induced by $ d $. Counterexamples are constructed when any of the hypotheses is absent. The behavior of the iterates of $ d_0 $ (the metrics $ d_0^n $ inductively defined as $ (d_0^{n-1})_0 $) is also considered.

math.MG↗
arXiv2015

Components of spaces of curves with constrained curvature on flat surfaces

Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval, in terms of all parameters involved. Many topological properties of these spaces are investigated. Some conjectures of L. E. Dubins are proved.

math.GT↗
arXiv2013

Homotopies of Curves on the 2-Sphere with Geodesic Curvature in a Prescribed Interval

Let $C_{k_1}^{k_2}$ denote the set of all closed curves of class $C^r$ on the sphere $S^2$ whose geodesic curvatures are restricted to lie in $(k_1,k_2)$, furnished with the $C^r$ topology (for some $r >= 2$ and possibly infinite $k_1 < k_2$). In 1970, J. Little proved that the space $C_0^{+\infty}$ of closed curves having positive geodesic curvature has three connected components. Let $r_i = arccot k_i$ (i = 1, 2). We show that $C_{k_1}^{k_2}$ has n connected components $C_1, ..., C_n$, where n is the greatest integer smaller than or equal to $π/(r_1-r_2) + 1$, and $C_j$ contains circles traversed j times ($1 <= j <= n$). The component $C_{n-1}$ also contains circles traversed $(n-1) + 2m$ times, and $C_n$ also contains circles traversed $n + 2m$ times, for any natural number m. In addition, each of $C_1, ..., C_{n-2}$ is homotopy equivalent to $SO_3$ ($n >= 3$). A simple characterization of the components in terms of the properties of a curve and a proof that $C_{k_1}^{k_2}$ is homeomorphic to $C_{k'_1}^{k'_2}$ whenever $r_1 - r_2 = r'_1 - r'_2$ ($r'_i = arccot k'_i$) are also presented.

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