Centroaffine differential geometry and its relations to horizontal submanifolds

Luc Vrancken

Displaying similar documents to “Centroaffine differential geometry and its relations to horizontal submanifolds”

The natural operators of general affine connections into general affine connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We reduce the problem of describing all $ℳ f_{m}$ -natural operators transforming general affine connections on $m$ -manifolds into general affine ones to the known description of all $G L (𝐑^{m})$ -invariant maps $𝐑^{m *} \otimes 𝐑^{m} \to \otimes^{k} 𝐑^{m *} \otimes \otimes^{k} 𝐑^{m}$ for $k = 1, 3$ .

On strongly affine extensions of commutative rings

Nabil Zeidi (2020)

Czechoslovak Mathematical Journal

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A ring extension $R \subseteq S$ is said to be strongly affine if each $R$ -subalgebra of $S$ is a finite-type $R$ -algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if $R$ is a quasi-local ring of finite dimension, then $R \subseteq S$ is integrally closed and strongly affine if and only if $R \subseteq S$ is a Prüfer extension (i.e. $(R, S)$ is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown....

Uniqueness of Cartesian Products of Compact Convex Sets

Zbigniew Lipecki, Viktor Losert, Jiří Spurný (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $X_{i}$ , i∈ I, and $Y_{j}$ , j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of $\prod_{i \in I} X_{i}$ onto $\prod_{j \in J} Y_{j}$ . We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_{i}$ onto $Y_{b (i)}$ , i∈ I.

Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Kevin G. Hare, Nikita Sidorov (2015)

Studia Mathematica

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Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if $| d e t M | \geq 2^{- 1 / d}$ , then the attractor $A_{M}$ has non-empty interior. We also consider the set $_{M}$ of points in $A_{M}$ which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of $_{M}$ is positive. For this special class the full description of $_{M}$ is given as well. This paper continues our...

The universal tropicalization and the Berkovich analytification

Jeffrey Giansiracusa, Noah Giansiracusa (2022)

Kybernetika

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Given an integral scheme $X$ over a non-archimedean valued field $k$ , we construct a universal closed embedding of $X$ into a $k$ -scheme equipped with a model over the field with one element $𝔽_{1}$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^{an}$ . Moreover, using the scheme-theoretic...

Poincaré inequalities and rigidity for actions on Banach spaces

Piotr Nowak (2015)

Journal of the European Mathematical Society

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The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan’s property (T) and is equivalent to the fact that $H^{1} (G, π) = 0$ for every isometric representation $π$ of $G$ on $X$ . The condition is expressed in terms of $p$ -Poincaré constants and we...

On the differential geometry of some classes of infinite dimensional manifolds

Maysam Maysami Sadr, Danial Bouzarjomehri Amnieh (2024)

Archivum Mathematicum

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Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space $Γ_{X}$ of any manifold $X$ . The name comes from the fact that various elements of the geometry of $Γ_{X}$ are constructed via lifting of the corresponding elements of the geometry of $X$ . In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to $X$ . In order to define...

Finiteness problems on Nash manifolds and Nash sets

José F. Fernando, José Manuel Gamboa, Jesús M. Ruiz (2014)

Journal of the European Mathematical Society

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We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$ . Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$ , (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_{1}, \dots, u_{m}) : U \to ℝ^{m}$ such that $U \cap X = {u_{1} \dots u_{r} = 0}$ , (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash...

Purity of level $m$ stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let $k$ be a field of characteristic $p > 0$ . Let $D_{m}$ be a ${BT}_{m}$ over $k$ (i.e., an $m$ -truncated Barsotti–Tate group over $k$ ). Let $S$ be a $k$ -scheme and let $X$ be a ${BT}_{m}$ over $S$ . Let $S_{D_{m}} (X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_{m}$ . If $p \geq 5$ , we show that $S_{D_{m}} (X)$ is pure in $S$ , i.e. the immersion $S_{D_{m}} (X) ↪ S$ is affine. For $p \in {2, 3}$ , we prove purity if $D_{m}$ satisfies a certain technical property depending only on its $p$ -torsion $D_{m} [p]$ . For $p \geq 5$ , we apply the developed techniques to show that...

Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents

Manuel del Pino, Fethi Mahmudi, Monica Musso (2014)

Journal of the European Mathematical Society

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Let $Ω$ be a bounded domain in $ℝ^{n}$ with smooth boundary $\partial Ω$ . We consider the equation $d^{2} Δ u - u + u^{\frac{n - k + 2}{n - k - 2}} = 0 in Ω$ , under zero Neumann boundary conditions, where $Ω$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$ -dimensional closed, embedded minimal submanifold $K$ of $\partial Ω$ , which is non-degenerate, and certain weighted average of sectional curvatures of $\partial Ω$ is positive along $K$ . Then we prove the existence of a sequence $d = d_{j} \to 0$ and a positive solution $u_{d}$ such that $d^{2} {| \nabla u_{d} |}^{2} ⇀ S δ_{K} as d \to 0$ in the sense of measures,...

$f$ -biminimal maps between Riemannian manifolds

Yan Zhao, Ximin Liu (2019)

Czechoslovak Mathematical Journal

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We give the definition of $f$ -biminimal submanifolds and derive the equation for $f$ -biminimal submanifolds. As an application, we give some examples of $f$ -biminimal manifolds. Finally, we consider $f$ -minimal hypersurfaces in the product space $ℝ^{n} \times 𝕊^{1} (a)$ and derive two rigidity theorems.

On transitive submanifolds of $𝒞^{2}$ and $𝒞^{3}$

Alois Švec (1973)

Czechoslovak Mathematical Journal

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Expansion in $S L_{d} (𝒪_{K} / I)$ , $I$ square-free

Péter P. Varjú (2012)

Journal of the European Mathematical Society

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Let $S$ be a fixed symmetric finite subset of $S L_{d} (𝒪_{K})$ that generates a Zariski dense subgroup of $S L_{d} (𝒪_{K})$ when we consider it as an algebraic group over $m a t h b b Q$ by restriction of scalars. We prove that the Cayley graphs of $S L_{d} (𝒪_{K} / I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of $𝒪_{K}$ if $d = 2$ and $K$ is an arbitrary numberfield, or if $d = 3$ and $K = ℚ$ .

Shells of monotone curves

Josef Mikeš, Karl Strambach (2015)

Czechoslovak Mathematical Journal

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We determine in $ℝ^{n}$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla$ for which any image of $C$ under a compact-free group $Ω$ of affinities containing the translation group is a geodesic with respect to $\nabla$ . Special attention is paid to the case that $Ω$ contains many dilatations or that $C$ is a curve in $ℝ^{3}$ . If $C$ is a curve in $ℝ^{3}$ and $Ω$ is the translation group then we calculate not only the components of the curvature and the Weyl...

Renormings of $c_{0}$ and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this paper is to show that for every Banach space $(X, ∥ \cdot ∥)$ containing asymptotically isometric copy of the space $c_{0}$ there is a bounded, closed and convex set $C \subset X$ with the Chebyshev radius $r (C) = 1$ such that for every $k \geq 1$ there exists a $k$ -contractive mapping $T : C \to C$ with $∥ x - T x ∥ > 1 - 1 / k$ for any $x \in C$ .