Displaying similar documents to “On the differential geometry of some classes of infinite dimensional manifolds”

On a new normalization for tractor covariant derivatives

Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan (2012)

Journal of the European Mathematical Society

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A regular normal parabolic geometry of type G / P on a manifold M gives rise to sequences D i of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ω on the corresponding tractor bundle V , where ω is the normal Cartan connection. The first operator D 0 in the sequence is overdetermined and it is well known that ω yields the prolongation of this operator in the homogeneous case M = G / P . Our first...

On a question of Schmidt and Summerer concerning 3 -systems

Johannes Schleischitz (2020)

Communications in Mathematics

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Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper 3 -system ( P 1 , P 2 , P 3 ) with the property ϕ ¯ 3 = 1 . In fact, our method generalizes to provide n -systems with ϕ ¯ n = 1 , for arbitrary n 3 . We visualize our constructions with graphics. We further present explicit examples of numbers ξ 1 , ... , ξ n - 1 that induce the n -systems in question.

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h i ( k , n ) be the i -th ordered configuration space of all distinct points H 1 , ... , H h in the Grassmannian G r ( k , n ) of k -dimensional subspaces of n , whose sum is a subspace of dimension i . We prove that h i ( k , n ) is (when non empty) a complex submanifold of G r ( k , n ) h of dimension i ( n - i ) + h k ( i - k ) and its fundamental group is trivial if i = m i n ( n , h k ) , h k n and n &gt; 2 and equal to the braid group of the sphere P 1 if n = 2 . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1 .

Involutivity degree of a distribution at superdensity points of its tangencies

Silvano Delladio (2021)

Archivum Mathematicum

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Let Φ 1 , ... , Φ k + 1 (with k 1 ) be vector fields of class C k in an open set U N + m , let 𝕄 be a N -dimensional C k submanifold of U and define 𝕋 : = { z 𝕄 : Φ 1 ( z ) , ... , Φ k + 1 ( z ) T z 𝕄 } where T z 𝕄 is the tangent space to 𝕄 at z . Then we expect the following property, which is obvious in the special case when z 0 is an interior point (relative to 𝕄 ) of 𝕋 : If z 0 𝕄 is a ( N + k ) -density point (relative to 𝕄 ) of 𝕋 then all the iterated Lie brackets of order less or equal to k Φ i 1 ( z 0 ) , [ Φ i 1 , Φ i 2 ] ( z 0 ) , [ [ Φ i 1 , Φ i 2 ] , Φ i 3 ] ( z 0 ) , ... ( h , i h k + 1 ) belong to T z 0 𝕄 . Such a property has been proved in [9] for k = 1 and its proof in the...

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi, Daniel Smania (2012)

Annales scientifiques de l'École Normale Supérieure

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We consider C 2 families t f t of  C 4 unimodal maps f t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure μ t of  f t depends differentiably on  t , as a distribution of order 1 . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of  μ t for a Benedicks-Carleson map f t , in terms of a single smooth function and the...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Characterization of cycle domains via Kobayashi hyperbolicity

Gregor Fels, Alan Huckleberry (2005)

Bulletin de la Société Mathématique de France

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A real form G of a complex semi-simple Lie group G has only finitely many orbits in any given G -flag manifold Z = G / Q . The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits D generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of D which, with very few well-understood exceptions, are parameterized by the Wolf cycle...

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

Selectors of discrete coarse spaces

Igor Protasov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Given a coarse space ( X , ) with the bornology of bounded subsets, we extend the coarse structure from X × X to the natural coarse structure on ( { } ) × ( { } ) and say that a macro-uniform mapping f : ( { } ) X (or f : [ X ] 2 X ) is a selector (or 2-selector) of ( X , ) if f ( A ) A for each A { } ( A [ X ] 2 , respectively). We prove that a discrete coarse space ( X , ) admits a selector if and only if ( X , ) admits a 2-selector if and only if there exists a linear order “ " on X such that the family of intervals { [ a , b ] : a , b X , a b } is a base for the bornology .

The general rigidity result for bundles of A -covelocities and A -jets

Jiří M. Tomáš (2017)

Czechoslovak Mathematical Journal

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Let M be an m -dimensional manifold and A = 𝔻 k r / I = N A a Weil algebra of height r . We prove that any A -covelocity T x A f T x A * M , x M is determined by its values over arbitrary max { width A , m } regular and under the first jet projection linearly independent elements of T x A M . Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T A * M T r * M without coordinate computations, which improves and generalizes the partial...