A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator

Maria Robaszewska

Displaying similar documents to “A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator”

Real hypersurfaces with an induced almost contact structure

Michał Szancer, Zuzanna Szancer (2009)

Colloquium Mathematicae

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We study real affine hypersurfaces $f : M \to ℂ^{n + 1}$ with an almost contact structure (φ,ξ,η) induced by any J-tangent transversal vector field. The main purpose of this paper is to show that if (φ,ξ,η) is metric relative to the second fundamental form then it is Sasakian and moreover f(M) is a piece of a hyperquadric in $ℝ^{2 n + 2}$ .

Real hypersurfaces with a special transversal vector field

Zuzanna Szancer (2012)

Annales Polonici Mathematici

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Real affine hypersurfaces of the complex space $ℂ^{n + 1}$ are studied. Some properties of the structure determined by a J-tangent transversal vector field are proved. Moreover, some generalizations of the results obtained by V. Cruceanu are given.

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 $k = 1, 3$ .

Complete noncompact submanifolds with flat normal bundle

Hai-Ping Fu (2016)

Annales Polonici Mathematici

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Let Mⁿ (n ≥ 3) be an n-dimensional complete super stable minimal submanifold in $ℝ^{n + p}$ with flat normal bundle. We prove that if the second fundamental form A of M satisfies $\int_{M} i {| A |}^{α} < \infty$ , where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and $\int_{M} {| A |}^{d} < \infty$ , d = 1,3, then M is an affine n-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite $L^{α}$ -norm curvature in ℝ⁷ are considered.

Centroaffine differential geometry and its relations to horizontal submanifolds

Luc Vrancken (2002)

Banach Center Publications

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We relate centroaffine immersions $f : M ⁿ \to ℝ^{n + 1}$ to horizontal immersions g of Mⁿ into $S_{n + 1}^{2 n + 1} (1)$ or $H_{n}^{2 n + 1} (- 1)$ . We also show that f is an equiaffine sphere, i.e. the centroaffine normal is a constant multiple of the Blaschke normal, if and only if g is minimal.

Integral representations for solutions of exponential Gauß-Manin systems

Marco Hien, Céline Roucairol (2008)

Bulletin de la Société Mathématique de France

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Let $f, g : U \to 𝔸^{1}$ be two regular functions from the smooth affine complex variety $U$ to the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system $𝒪_{U} e^{g}$ with respect to $f$ . We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.

On the Hausdorff dimension of certain self-affine sets

Abercrombie Alex G.., Nair R. (2002)

Studia Mathematica

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A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $Φ (S) = ⋃_{1 \leq j \leq t} ϕ_{j} (S)$ . If Φ(S) ⊂ S let $E_{Φ} (S)$ denote $⋂_{k \geq 0} Φ^{k} (S)$ . For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{Φ} (S)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also...

On Some Family of Contractible Hypersurfaces in $C^{4}$

S. Kaliman, L. Makar-Limanov (1993)

Cours de l'institut Fourier

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Sufficient conditions for the spectrality of self-affine measures with prime determinant

Jian-Lin Li (2014)

Studia Mathematica

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Let $μ_{M, D}$ be a self-affine measure associated with an expanding matrix M and a finite digit set D. We study the spectrality of $μ_{M, D}$ when |det(M)| = |D| = p is a prime. We obtain several new sufficient conditions on M and D for $μ_{M, D}$ to be a spectral measure with lattice spectrum. As an application, we present some properties of the digit sets of integral self-affine tiles, which are connected with a conjecture of Lagarias and Wang.

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.

On some properties of induced almost contact structures

Zuzanna Szancer (2015)

Annales Polonici Mathematici

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Real affine hypersurfaces of the complex space $ℂ^{n + 1}$ with a J-tangent transversal vector field and an induced almost contact structure (φ,ξ,η) are studied. Some properties of the induced almost contact structures are proved. In particular, we prove some properties of the induced structure when the distribution is involutive. Some constraints on a shape operator when the induced almost contact structure is either normal or ξ-invariant are also given.

A characterization of a certain real hypersurface of type $(A_{2})$ in a complex projective space

Byung Hak Kim, In-Bae Kim, Sadahiro Maeda (2017)

Czechoslovak Mathematical Journal

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In the class of real hypersurfaces $M^{2 n - 1}$ isometrically immersed into a nonflat complex space form ${\tilde{M}}_{n} (c)$ of constant holomorphic sectional curvature $c$ $(\neq 0)$ which is either a complex projective space $ℂ P^{n} (c)$ or a complex hyperbolic space $ℂ H^{n} (c)$ according as $c > 0$ or $c < 0$ , there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds....

New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino, Henrique F. de Lima (2015)

Archivum Mathematicum

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In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space $ℍ^{n + 1}$ , that is, complete hypersurfaces of $ℍ^{n + 1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R = a H + b$ for some $a$ , $b \in ℝ$ . In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $ℍ^{n + 1}$ . Furthermore,...

Distances to spaces of affine Baire-one functions

Jiří Spurný (2010)

Studia Mathematica

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Let E be a Banach space and let $ℬ ₁ (B_{E *})$ and $₁ (B_{E *})$ denote the space of all Baire-one and affine Baire-one functions on the dual unit ball $B_{E *}$ , respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between $d i s t (f, ℬ ₁ (B_{E *}))$ and $d i s t (f, ₁ (B_{E *}))$ , where f is an affine function on $B_{E *}$ . If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.

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...

Horizontal lift of symmetric connections to the bundle of volume forms $𝒱$

Anna Gasior (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we present the horizontal lift of a symmetric affine connection with respect to another affine connection to the bundle of volume forms $𝒱$ and give formulas for its curvature tensor, Ricci tensor and the scalar curvature. Next, we give some properties of the horizontally lifted vector fields and certain infinitesimal transformations. At the end, we consider some substructures of a $F (3, 1)$ -structure on $𝒱$ .

Self-affine measures and vector-valued representations

Qi-Rong Deng, Xing-Gang He, Ka-Sing Lau (2008)

Studia Mathematica

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Let A be a d × d integral expanding matrix and let $S_{j} (x) = A^{- 1} (x + d_{j})$ for some $d_{j} \in ℤ^{d}$ , j = 1,...,m. The iterated function system (IFS) ${S_{j}}_{j = 1}^{m}$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof...

A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Zinovy Reichstein, Angelo Vistoli (2013)

Journal of the European Mathematical Society

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We compute the essential dimension of the functors Forms $_{n, d}$ and Hypersurf $_{n, d}$ of equivalence classes of homogeneous polynomials in $n$ variables and hypersurfaces in $ℙ^{n - 1}$ , respectively, over any base field $k$ of characteristic $0$ . Here two polynomials (or hypersurfaces) over $K$ are considered equivalent if they are related by a linear change of coordinates with coefficients in $K$ . Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application...

On two problems studied by A. Ambrosetti

David Arcoya, José Carmona (2006)

Journal of the European Mathematical Society

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We study the Ambrosetti–Prodi and Ambrosetti–Rabinowitz problems.We prove for the first one the existence of a continuum of solutions with shape of a reflected $C$ ( $\supset$ -shape). Next, we show that there is a relationship between these two problems.

The tame automorphism group of an affine quadric threefold acting on a square complex

Cinzia Bisi, Jean-Philippe Furter, Stéphane Lamy (2014)

Journal de l’École polytechnique — Mathématiques

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We study the group $Tame ({SL}_{2})$ of tame automorphisms of a smooth affine $3$ -dimensional quadric, which we can view as the underlying variety of ${SL}_{2} (ℂ)$ . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $CAT (0)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $Tame ({SL}_{2})$ is linearizable, and that $Tame ({SL}_{2})$ satisfies the Tits alternative.

On convex hypersurfaces in $E^{n + 1}$

Thomas Hasanis (1984)

Colloquium Mathematicae

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Hypersurfaces with singular locus a plane curve and transversal type $A_{1}$

Dirk Siersma (1988)

Banach Center Publications

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Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

Fabio Nicola (2010)

Studia Mathematica

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We study Fourier integral operators of Hörmander’s type acting on the spaces $ℱ L^{p} {(ℝ^{d})}_{c o m p}$ , 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in $L^{p}$ . We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on $ℱ L^{p} {(ℝ^{d})}_{c o m p}$ if the mapping $x \mapsto \nabla_{x} Φ (x, η)$ is constant on the fibres, of codimension r,...

Spacelike intersection curve of three spacelike hypersurfaces in $E_{1}^{4}$

B. Uyar Duldul, M. Caliskan (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space $E_{1}^{4}$ .

Generalizing the Johnson-Lindenstrauss lemma to k-dimensional affine subspaces

Alon Dmitriyuk, Yehoram Gordon (2009)

Studia Mathematica

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Let ε > 0 and 1 ≤ k ≤ n and let ${W_{l}}_{l = 1}^{p}$ be affine subspaces of ℝⁿ, each of dimension at most k. Let $m = O (ε^{- 2} (k + l o g p))$ if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map $H : ℝ ⁿ \to ℝ^{m}$ such that for all 1 ≤ l ≤ p and $x, y \in W_{l}$ we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces...

$J$ -holomorphic discs and real analytic hypersurfaces

William Alexandre, Emmanuel Mazzilli (2014)

Annales de l’institut Fourier

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We give in $ℝ^{6}$ a real analytic almost complex structure $J$ , a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$ , such that there is no germ of $J$ -holomorphic disc $γ$ included in $M$ with $γ (0) = 0$ and $\frac{\partial γ}{\partial x} (0) = v$ , although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$ , we give sufficient conditions under which there exists such a germ of disc.

On some ergodic properties for continuous and affine functions

Charles J. K. Batty (1978)

Annales de l'institut Fourier

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Two problems posed by Choquet and Foias are solved: (i) Let $T$ be a positive linear operator on the space $C (X)$ of continuous real-valued functions on a compact Hausdorff space $X$ . It is shown that if $n^{- 1} \sum_{r = 0}^{n - 1} T^{r} 1$ converges pointwise to a continuous limit, then the convergence is uniform on $X$ . (ii) An example is given of a Choquet simplex $K$ and a positive linear operator $T$ on the space $A (K)$ of continuous affine real-valued functions on $K$ , such that $\inf {(T^{n} 1) (x) : n \geq} < 1$ for each...

Digit sets of integral self-affine tiles with prime determinant

Jian-Lin Li (2006)

Studia Mathematica

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Let M ∈ Mₙ(ℤ) be expanding such that |det(M)| = p is a prime and pℤⁿ ⊈ M²(ℤⁿ). Let D ⊂ ℤⁿ be a finite set with |D| = |det(M)|. Suppose the attractor T(M,D) of the iterated function system ${ϕ_{d} (x) = M^{- 1} (x + d)}_{d \in D}$ has positive Lebesgue measure. We prove that (i) if D ⊈ M(ℤⁿ), then D is a complete set of coset representatives of ℤⁿ/M(ℤⁿ); (ii) if D ⊆ M(ℤⁿ), then there exists a positive integer γ such that $D = M^{γ} D ₀$ , where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results...