Isotropic almost complex structures and harmonic unit vector fields

Amir Baghban; Esmaeil Abedi

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Displaying similar documents to “Isotropic almost complex structures and harmonic unit vector fields”

Deformations of Metrics and Biharmonic Maps

Aicha Benkartab, Ahmed Mohammed Cherif (2020)

Communications in Mathematics

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We construct biharmonic non-harmonic maps between Riemannian manifolds $(M, g)$ and $(N, h)$ by first making the ansatz that $ϕ : (M, g) \to (N, h)$ be a harmonic map and then deforming the metric on $N$ by ${\tilde{h}}_{α} = α h + (1 - α) d f \otimes d f$ to render $ϕ$ biharmonic, where $f$ is a smooth function with gradient of constant norm on $(N, h)$ and $α \in (0, 1)$ . We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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We study local reflections $\phi_{\sigma}$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if $\phi_{\sigma}$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if $\phi_{\sigma}$ is harmonic for all geodesies $\sigma$ .

Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Landau's theorem for p-harmonic mappings in several variables

Sh. Chen, S. Ponnusamy, X. Wang (2012)

Annales Polonici Mathematici

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A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation $Δ^{p} f = 0$ , where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f : Ω \to ℂ^{m}$ is said to be p-harmonic in Ω if each component function $f_{i}$ (i∈ 1,...,m) of $f = (f ₁, . . ., f_{m})$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

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If $(M, \nabla)$ is a manifold with a symmetric linear connection, then $T^{*} M$ can be endowed with the natural Riemann extension $\bar{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\bar{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $𝒫$ on $(T^{*} M, \bar{g})$ and prove that $𝒫$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\bar{g}$ reduces...

Complete Riemannian manifolds admitting a pair of Einstein-Weyl structures

Amalendu Ghosh (2016)

Mathematica Bohemica

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We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g, \pm ω)$ with constant scalar curvature is either Einstein, or the dual field of $ω$ is Killing. Next, let $(M^{n}, g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g, \pm ω)$ . Then the Einstein-Weyl vector field $E$ (dual to the $1$ -form $ω$ ) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.

Collapse of warped submersions

Szymon M. Walczak (2006)

Annales Polonici Mathematici

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We generalize the concept of warped manifold to Riemannian submersions π: M → B between two compact Riemannian manifolds $(M, g_{M})$ and $(B, g_{B})$ in the following way. If f: B → (0,∞) is a smooth function on B which is extended to a function f̂ = f ∘ π constant along the fibres of π then we define a new metric $g_{f}$ on M by $g_{f} |_{\times} \equiv g_{M} |_{\times}, g_{f} {|_{\times T M ̂} \equiv f ̂ ² g_{M} |}_{\times T M ̂}$ , where and denote the bundles of horizontal and vertical vectors. The manifold $(M, g_{f})$ obtained that way is called a warped submersion. The function f is called a warping function. We show...

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek, Włodzimierz Mikulski (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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If $(M, g)$ is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism $T M \tilde{=} T^{*} M$ given by $v \to g (v, -)$ between the tangent $T M$ and the cotangent $T^{*} M$ bundles of $M$ . In the present note, we generalize this isomorphism to the one $T^{(r)} M \tilde{=} T^{r *} M$ between the $r$ -th order vector tangent $T^{(r)} M = {(J^{r} {(M, R)}_{0})}^{*}$ and the $r$ -th order cotangent $T^{r *} M = J^{r} {(M, R)}_{0}$ bundles of $M$ . Next, we describe all base preserving vector bundle maps $C_{M} (g) : T^{(r)} M \to T^{r *} M$ depending on a Riemannian metric $g$ in terms of natural (in $g$ ) tensor fields on $M$ .

Some characterizations of harmonic Bloch and Besov spaces

Xi Fu, Bowen Lu (2016)

Czechoslovak Mathematical Journal

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The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic $ω$ - $α$ -Bloch space and characterize it in terms of $ω ((1 - {| x |}^{2})^{β} {(1 - | y |}^{2})^{α - β}) | \frac{f (x) - f (y)}{x - y} |$ and $ω ((1 - {| x |}^{2})^{β} {(1 - | y |}^{2})^{α - β}) | \frac{f (x) - f (y)}{| x | y - x^{'}} |$ where $ω$ is a majorant. Similar results are extended to harmonic little $ω$ - $α$ -Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).

On the dimension of $p$ -harmonic measure in space

John L. Lewis, Kaj Nyström, Andrew Vogel (2013)

Journal of the European Mathematical Society

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Let $Ω \subset ℝ^{n}, n \geq 3$ , and let $p, 1 < p < \infty, p \neq 2$ , be given. In this paper we study the dimension of $p$ -harmonic measures that arise from non-negative solutions to the $p$ -Laplace equation, vanishing on a portion of $\partial Ω$ , in the setting of $δ$ -Reifenberg flat domains. We prove, for $p \geq n$ , that there exists $\tilde{δ} = \tilde{δ} (p, n) > 0$ small such that if $Ω$ is a $δ$ -Reifenberg flat domain with $δ < \tilde{δ}$ , then $p$ -harmonic measure is concentrated on a set of $σ$ -finite $H^{n - 1}$ -measure. We prove, for $p \geq n$ , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$ -harmonic...

On a question of T. Sheil-Small regarding valency of harmonic maps

Daoud Bshouty, Abdallah Lyzzaik (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form $f (e^{i t}) = e^{i φ (t)}$ , $0 \leq t \leq 2 π$ where $φ$ is a continuously non-decreasing function that satisfies $φ (2 π) - φ (0) = 2 N π$ , assume every value finitely many times in the disc?

Special sets of reals and weak forms of normality on Isbell--Mrówka spaces

Vinicius de Oliveira Rodrigues, Victor dos Santos Ronchim, Paul J. Szeptycki (2023)

Commentationes Mathematicae Universitatis Carolinae

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We recall some classical results relating normality and some natural weakenings of normality in $Ψ$ -spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$ -sets, $λ$ -sets and $σ$ -sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being $ℵ_{0}$ -separated. This new class fits between $λ$ -sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $𝒜$ being...

A Weighted Eigenvalue Problems Driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -Biharmonic Operators

Mohamed Laghzal, Abdelouahed El Khalil, Abdelfattah Touzani (2021)

Communications in Mathematics

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The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -biharmonic operators $\begin{matrix} Δ_{p (x)}^{2} u - Δ_{p (x)} u = λ w (x) {| u |}^{q (x) - 2} u in Ω, \\ u \in W^{2, p (\cdot)} (Ω) \cap W_{0}^{1, p (\cdot)} (Ω), \end{matrix}$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p (\cdot)} (Ω)$ and $W^{m, p (\cdot)} (Ω)$ .

Induced almost continuous functions on hyperspaces

Alejandro Illanes (2006)

Colloquium Mathematicae

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For a metric continuum X, let C(X) (resp., $2^{X}$ ) be the hyperspace of subcontinua (resp., nonempty closed subsets) of X. Let f: X → Y be an almost continuous function. Let C(f): C(X) → C(Y) and $2^{f} : 2^{X} \to 2^{Y}$ be the induced functions given by $C (f) (A) = c l_{Y} (f (A))$ and $2^{f} (A) = c l_{Y} (f (A))$ . In this paper, we prove that: • If $2^{f}$ is almost continuous, then f is continuous. • If C(f) is almost continuous and X is locally connected, then f is continuous. • If X is not locally connected, then there exists an almost continuous function f: X → [0,1]...

On the integral representation of finely superharmonic functions

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri (2019)

Commentationes Mathematicae Universitatis Carolinae

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In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $𝒫$ -harmonic space $Ω$ with countable base of open subsets and satisfying the axiom $D$ . When $Ω$ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$ . As an application of the integral representation we extend to the cone...

On $G$ -sets and isospectrality

Ori Parzanchevski (2013)

Annales de l’institut Fourier

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We study finite $G$ -sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: If $M$ is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then $M$ has isospectral non-isometric covers.

Asymmetric tie-points and almost clopen subsets of $ℕ^{*}$

Alan S. Dow, Saharon Shelah (2018)

Commentationes Mathematicae Universitatis Carolinae

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A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of $ℕ^{*}$ . One especially important application, due to Veličković, was to the existence of nontrivial involutions on $ℕ^{*}$ . A tie-point of $ℕ^{*}$ has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost...

Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ${(h_{k})}_{k \geq 0}$ be the Haar system on [0,1]. We show that for any vectors $a_{k}$ from a separable Hilbert space and any $ε_{k} \in [- 1, 1]$ , k = 0,1,2,..., we have the sharp inequality $| | \sum_{k = 0}^{n} ε_{k} a_{k} h_{k} {| |}_{W ([0, 1])} \leq 2 | | \sum_{k = 0}^{n} a_{k} h_{k} {| |}_{L^{\infty} ([0, 1])}$ , n = 0,1,2,..., where W([0,1]) is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${| | Y | |}_{W (Ω)} {\leq 2 | | X | |}_{L^{\infty} (Ω)}$ , where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.