Tenseness of Riemannian flows

Hiraku Nozawa; José Ignacio Royo Prieto

Displaying similar documents to “Tenseness of Riemannian flows”

A Weitzenbôck formula for the second fundamental form of a Riemannian foliation

Paolo Piccinni (1984)

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

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Si considera la seconda forma fondamentale $\alpha$ di foliazioni su varietà riemanniane e si ottiene una formula per il laplaciano $\nabla^{2}\alpha$ - Se ne deducono alcune implicazioni per foliazioni su varietà a curvatura costante.

On some properties of a Riemannian space $V_{4}$ with constant scalar curvature

W. Slósarska, Z. Żekanowski (1972)

Colloquium Mathematicae

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Riemannian semisymmetric almost Kenmotsu manifolds and nullity distributions

Yaning Wang, Ximin Liu (2014)

Annales Polonici Mathematici

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We consider an almost Kenmotsu manifold $M^{2 n + 1}$ with the characteristic vector field ξ belonging to the (k,μ)’-nullity distribution and h’ ≠ 0 and we prove that $M^{2 n + 1}$ is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, provided that $M^{2 n + 1}$ is ξ-Riemannian-semisymmetric. Moreover, if $M^{2 n + 1}$ is a ξ-Riemannian-semisymmetric almost Kenmotsu manifold such that ξ belongs to the (k,μ)-nullity distribution, we prove...

A Weitzenbôck formula for the second fundamental form of a Riemannian foliation

Paolo Piccinni (1984)

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

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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...

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.

Groups of $C^{r, s}$ -diffeomorphisms related to a foliation

Jacek Lech, Tomasz Rybicki (2007)

Banach Center Publications

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The notion of a $C^{r, s}$ -diffeomorphism related to a foliation is introduced. A perfectness theorem for the group of $C^{r, s}$ -diffeomorphisms is proved. A remark on $C^{n + 1}$ -diffeomorphisms is given.

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.

Generalized gradient flow and singularities of the Riemannian distance function

Piermarco Cannarsa (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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Significant information about the topology of a bounded domain $Ω$ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial Ω}$ , from the boundary of $Ω$ . We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_{\partial Ω}$ , as well as applications to homotopy equivalence.

Minimal, rigid foliations by curves on $ℂ ℙ^{n}$

Frank Loray, Julio C. Rebelo (2003)

Journal of the European Mathematical Society

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We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space $ℂ ℙ^{n}$ for every dimension $n \geq 2$ and every degree $d \geq 2$ . Precisely, we construct a foliation $ℱ$ which is induced by a homogeneous vector field of degree $d$ , has a finite singular set and all the regular leaves are dense in the whole of $ℂ ℙ^{n}$ . Moreover, $ℱ$ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if $ℱ$ is conjugate to another holomorphic...

Global existence of solutions to Schrödinger equations on compact riemannian manifolds below $H^{1}$

Sijia Zhong (2010)

Bulletin de la Société Mathématique de France

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In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. $s < 1$ , under some bilinear Strichartz assumption. We will find some $\tilde{s} < 1$ , such that the solution is global for $s > \tilde{s}$ .

Riemannian geometries on spaces of plane curves

Peter W. Michor, David Mumford (2006)

Journal of the European Mathematical Society

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We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from $S^{1}$ to the plane modulo the group of diffeomorphisms of $S^{1}$ , acting as reparametrizations. In particular we investigate the metric, for a constant $A > 0$ , $G_{c}^{A} (h, k) : = \int_{S^{1}} (1 + A κ_{c} {(θ)}^{2}) 〈 h (θ), k (θ) 〉 | c^{'} (θ) | d θ$ where $κ_{c}$ is the curvature of the curve $c$ and $h$ , $k$ are normal vector fields to $c$ . The term $A κ^{2}$ is a sort of geometric Tikhonov regularization because, for $A = 0$ , the geodesic distance between any two distinct curves is 0, while...

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.

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 $\sigma$ .

On $2p$ -dimensional Riemannian manifolds with positive scalar curvature

Domenico Perrone (1984)

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

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In questo lavoro si danno alcuni risultati sugli spettri degli operatori di Laplace per varietà Riemanniane compatte con curvatura scalare positiva e di dimensione $2p$ . Ad essi si aggiunge una osservazione riguardante la congettura di Yamabe.

Growth of a primitive of a differential form

Jean-Claude Sikorav (2001)

Bulletin de la Société Mathématique de France

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For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function $f$ , by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if $M$ has bounded geometry. For a volume form, it suffices to have the inequality ( $| Ω | \leq \int_{\partial Ω} f d σ$ for every compact domain $Ω \subset M$ ). This implies in particular the “well-known” result that if $M$ is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then...

Twisted cotangent sheaves and a Kobayashi-Ochiai theorem for foliations

Andreas Höring (2014)

Annales de l’institut Fourier

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Let $X$ be a normal projective variety, and let $A$ be an ample Cartier divisor on $X$ . Suppose that $X$ is not the projective space. We prove that the twisted cotangent sheaf $Ω_{X} \otimes A$ is generically nef with respect to the polarisation $A$ . As an application we prove a Kobayashi-Ochiai theorem for foliations: if $ℱ ⊊ T_{X}$ is a foliation such that $det ℱ \equiv i_{ℱ} A$ , then $i_{ℱ}$ is at most the rank of $ℱ$ .

Foliations by curves with curves as singularities

M. Corrêa Jr, A. Fernández-Pérez, G. Nonato Costa, R. Vidal Martins (2014)

Annales de l’institut Fourier

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Let $ℱ$ be a holomorphic one-dimensional foliation on $ℙ^{n}$ such that the components of its singular locus $Σ$ are curves $C_{i}$ and points $p_{j}$ . We determine the number of $p_{j}$ , counted with multiplicities, in terms of invariants of $ℱ$ and $C_{i}$ , assuming that $ℱ$ is special along the $C_{i}$ . Allowing just one nonzero dimensional component on $Σ$ , we also prove results on when the foliation happens to be determined by its singular locus.

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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Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek (2015)

Czechoslovak Mathematical Journal

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Let $(M, g)$ be a 4-dimensional Einstein Riemannian manifold. At each point $p$ of $M$ , the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor $R$ at $p$ . In this basis, up to standard symmetries and antisymmetries, just $5$ components of the curvature tensor $R$ are nonzero. For the space of constant curvature, the group $O (4)$ acts as a transformation group between ST bases at $T_{p} M$ and for the so-called 2-stein curvature tensors, the group $Sp (1) \subset SO (4)$ acts as a transformation...

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 $M$ .

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

Nguyen Ngoc Khanh (2016)

Archivum Mathematicum

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In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M, g)$ for the following general heat equation $u_{t} = Δ_{V} u + a u log u + b u$ where $a$ is a constant and $b$ is a differentiable function defined on $M \times [0, \infty)$ . We suppose that the Bakry-Émery curvature and the $N$ -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently. ...