Open Subsets of LF-spaces

Kotaro Mine; Katsuro Sakai

Displaying similar documents to “Open Subsets of LF-spaces”

The Killing Tensors on an $n$ -dimensional Manifold with $S L (n,)$ -structure

Sergey E. Stepanov, Irina I. Tsyganok, Marina B. Khripunova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$ -dimensional differentiable manifold $M$ endowed with an equiaffine $S L (n,)$ -structure and discuss possible applications of obtained results in Riemannian geometry.

Strong density for higher order Sobolev spaces into compact manifolds

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2015)

Journal of the European Mathematical Society

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Given a compact manifold $N^{n}$ , an integer $k \in ℕ_{*}$ and an exponent $1 \leq p < \infty$ , we prove that the class $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ of smooth maps on the cube with values into $N^{n}$ is dense with respect to the strong topology in the Sobolev space $W^{k, p} (Q^{m}; N^{n})$ when the homotopy group $π_{⌊ k p ⌋} (N^{n})$ of order $⌊ k p ⌋$ is trivial. We also prove density of maps that are smooth except for a set of dimension $m - ⌊ k p ⌋ - 1$ , without any restriction on the homotopy group of $N^{n}$ .

Remarks on the behaviour of higher-order derivations on the gluing of differential spaces

Krzysztof Drachal (2015)

Czechoslovak Mathematical Journal

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This paper is about some geometric properties of the gluing of order $k$ in the category of Sikorski differential spaces, where $k$ is assumed to be an arbitrary natural number. Differential spaces are one of possible generalizations of the concept of an infinitely differentiable manifold. It is known that in many (very important) mathematical models, the manifold structure breaks down. Therefore it is important to introduce a more general concept. In this paper, in particular, the behaviour...

Local density of diffeomorphisms with large centralizers

Christian Bonatti, Sylvain Crovisier, Gioia M. Vago, Amie Wilkinson (2008)

Annales scientifiques de l'École Normale Supérieure

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Given any compact manifold $M$ , we construct a non-empty open subset $𝒪$ of the space ${Diff}^{1} (M)$ of $C^{1}$ -diffeomorphisms and a dense subset $𝒟 \subset 𝒪$ such that the centralizer of every diffeomorphism in $𝒟$ is uncountable, hence non-trivial.

$η$ -Ricci Solitons on $η$ -Einstein ${(L C S)}_{n}$ -Manifolds

Shyamal Kumar Hui, Debabrata Chakraborty (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The object of the present paper is to study $η$ -Ricci solitons on $η$ -Einstein ${(L C S)}_{n}$ -manifolds. It is shown that if $ξ$ is a recurrent torse forming $η$ -Ricci soliton on an $η$ -Einstein ${(L C S)}_{n}$ -manifold then $ξ$ is (i) concurrent and (ii) Killing vector field.

The vertical prolongation of the projectable connections

Anna Bednarska (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We prove that any first order $ℱ_{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operator transforming projectable general connections on an $(m_{1}, m_{2}, n_{1}, n_{2})$ -dimensional fibred-fibred manifold $p = (p, p) : (p_{Y} : Y \to Y) \to (p_{M} : M \to M)$ into general connections on the vertical prolongation $V Y \to M$ of $p : Y \to M$ is the restriction of the (rather well-known) vertical prolongation operator $𝒱$ lifting general connections $\bar{Γ}$ on a fibred manifold $Y \to M$ into $𝒱 \bar{Γ}$ (the vertical prolongation of $\bar{Γ}$ ) on $V Y \to M$ .

Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals

Shuzo Izumi (2015)

Annales Polonici Mathematici

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Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection $_{b}$ from the local ring $_{n, b}$ onto the space $Z_{b}$ of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and $_{b}$ induces the structure of an Artinian algebra on $Z_{b}$ . In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈...

On canonical constructions on connections

Jan Kurek, Włodzimierz Mikulski (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We study how a projectable general connection $Γ$ in a 2-fibred manifold $Y^{2} \to Y^{1} \to Y^{0}$ and a general vertical connection $Θ$ in $Y^{2} \to Y^{1} \to Y^{0}$ induce a general connection $A (Γ, Θ)$ in $Y^{2} \to Y^{1}$ .

A density version of the Carlson–Simpson theorem

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros (2014)

Journal of the European Mathematical Society

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We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k \geq 2$ and every set $A$ of words over $k$ satisfying $\lim \sup_{n \to \infty} | A \cap {[k]}^{n} | / k^{n} > 0$ there exist a word $c$ over $k$ and a sequence $(w_{n})$ of left variable words over $k$ such that the set $c \cup {c^{} w_{0} {(a_{0})}^{} . . .^{} w_{n} (a_{n}) : n \in ℕ and a_{0}, . . ., a_{n} \in [k]}$ is contained in $A$ . While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine, Katsuro Sakai, Masato Yaguchi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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By Fin(X) (resp. $F i n^{k} (X)$ ), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂(τ) be the Hilbert space with weight τ and $ℓ ₂^{f} (τ)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E = ℓ ₂^{f} (τ)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes $_{α} (τ)$ and $_{α} (τ)$ of weight ≤ τ (α > 0) then Fin(E) and each $F i n^{k} (E)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X)...

Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends

Tatsuhiko Yagasaki (2007)

Fundamenta Mathematicae

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Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and $ℋ_{E} (M, ω)$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{ω} : ℋ_{E} (M, ω) \to (M, ω)$ , which measures for each $h \in ℋ_{E} (M, ω)$ the mass flow toward ends induced by h. We show that the...

On strong $(N, p, α)$ sumability of infinite series III.

R. Sinha (1973)

Matematički Vesnik

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Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $ℓ$ simply connected manifolds when $s \geq 1$

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2013)

Confluentes Mathematici

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Given a compact manifold $N^{n} \subset ℝ^{ν}$ and real numbers $s \geq 1$ and $1 \leq p < \infty$ , we prove that the class $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ of smooth maps on the cube with values into $N^{n}$ is strongly dense in the fractional Sobolev space $W^{s, p} (Q^{m}; N^{n})$ when $N^{n}$ is $⌊ s p ⌋$ simply connected. For $s p$ integer, we prove weak sequential density of $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ when $N^{n}$ is $s p - 1$ simply connected. The proofs are based on the existence of a retraction of $ℝ^{ν}$ onto $N^{n}$ except for a small subset of $N^{n}$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, s p}$ .

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$ . We consider the image $𝒩_{1} (D, X)$ of $𝒩_{1} (D)$ in $𝒩_{1} (X)$ under the natural push-forward of $1$ -cycles. We show that $ρ_{X} - ρ_{D} \leq codim 𝒩_{1} (D, X) \leq 8$ . Moreover if $codim 𝒩_{1} (D, X) \geq 3$ , then either $X ≅ S \times T$ where $S$ is a Del Pezzo surface, or $codim 𝒩_{1} (D, X) = 3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that $ρ_{X} - ρ_{T} = 4$ .

Holomorphic line bundles and divisors on a domain of a Stein manifold

Makoto Abe (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^{k} (D, 𝒪) = 0$ for $2 \leq k \leq n - 1$ . We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$ .

A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

Annales Polonici Mathematici

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Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let $π_{V}$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an $^{m}$ -measurable subset of ℝⁿ with $^{m} (A) < \infty$ . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V, v) | V \in G (n, m), v \in V$ such that, for all P ∈ A, one has $^{m (n - m)} (V \in G (n, m) | (V, π_{V} (P)) \in Z) > 0$ . One can replace “for all P ∈ A” by “for...

Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds

Stephen S.-T. Yau (2011)

Journal of the European Mathematical Society

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Let $X_{1}$ and $X_{2}$ be two compact strongly pseudoconvex CR manifolds of dimension $2 n - 1 \geq 5$ which bound complex varieties $V_{1}$ and $V_{2}$ with only isolated normal singularities in $ℂ^{N 1}$ and $ℂ^{N 2}$ respectively. Let $S_{1}$ and $S_{2}$ be the singular sets of $V_{1}$ and $V_{2}$ respectively and $S_{2}$ is nonempty. If $2 n - N_{2} - 1 \geq 1$ and the cardinality of $S_{1}$ is less than 2 times the cardinality of $S_{2}$ , then we prove that any non-constant CR morphism from $X_{1}$ to $X_{2}$ is necessarily a CR biholomorphism. On the other hand, let $X$ be a compact strongly pseudoconvex CR...

Effective finding of the solutions of the form $\sum_{l = 1}^{\infty} \frac{a_{l}}{x^{l}}$ for the differential equations of the finite and infinite order

K. Orlov (1980)

Matematički Vesnik

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Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

Daniel Greb, Christian Lehn, Sönke Rollenske (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a compact hyperkähler manifold containing a complex torus $L$ as a Lagrangian subvariety. Beauville posed the question whether $X$ admits a Lagrangian fibration with fibre $L$ . We show that this is indeed the case if $X$ is not projective. If $X$ is projective we find an almost holomorphic Lagrangian fibration with fibre $L$ under additional assumptions on the pair $(X, L)$ , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic...