Showing papers on "Reeb vector field published in 2018"
TL;DR: In this article, the authors consider a sub-Riemannian Laplacian with an oriented contact distribution and derive a Birkhoff normal form along this characteristic manifold, thus showing that all 3D contact structures are microlocally equivalent.
Abstract: This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasi-modes, trace formulae.
Even if hypoelliptic operators have been well studied from the point of view of PDE's, global geometrical and dynamical aspects have not been the subject of much attention. As we will see, already in the simplest case, the statements of the results in the sub-Riemannian setting are quite different from those in the Riemannian one.
Let us consider a sub-Riemannian (sR) metric on a closed three-dimensional manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We establish a Quantum Ergodicity (QE) theorem for the eigenfunctions of any associated sR Laplacian under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized Popp measure.
This is the first time that such a result is established for a hypoelliptic operator, whereas the usual Shnirelman theorem yields QE for the Laplace-Beltrami operator on a closed Riemannian manifold with ergodic geodesic flow.
To prove our theorem, we first establish a microlocal Weyl law, which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this characteristic manifold, thus showing that, in some sense, all 3D contact structures are microlocally equivalent. The quantum version of this normal form provides a useful microlocal factorization of the sR Laplacian. Using the normal form, the factorization and the ergodicity assumption, we finally establish a variance estimate, from which QE follows.
We also obtain a second result, which is valid without any ergodicity assumption: every Quantum Limit (QL) can be interpreted as the sum of two mutually singular measures: the first measure is supported on the unit cotangent bundle and is invariant under the sR geodesic flow, and the second measure is supported on the characteristic manifold of the sR Laplacian and is invariant under the lift of the Reeb flow. Moreover, we prove that the first measure is zero for most QL's.
28 citations
TL;DR: In this article, the authors use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein-Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone).
Abstract: Building on an idea laid out by Martelli, Sparks and Yau (2008), we use the Duistermaat–Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein–Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms, we prove they are all proper. Among consequences thereof we get that the Einstein–Hilbert functional attains its minimal value and each Sasaki cone possesses at least one Reeb vector field with vanishing transverse Futaki invariant.
16 citations
01 Jun 2018
TL;DR: In this paper, it was shown that a Ricci soliton whose soliton vector field is divergence-free and its soliton soliton is Killing can be reduced to Ricci Soliton if and only if the associated vector field has geodesic this paper.
Abstract: We show that a compact almost Ricci soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. Next we show that an almost Ricci soliton reduces to Ricci soliton if and only if the associated vector field is geodesic. Finally, we prove that a contact metric manifold is K-contact if and only if its Reeb vector field is geodesic.
13 citations
TL;DR: In this paper, the authors studied conformal slant submersions from cosymplectic manifolds onto Riemannian manifolds and obtained the geometries of the leaves of vertical distribution and horizontal distribution, including the integrability of the distributions, the geometry of foliations, some conditions related to total geodesicness, and harmonicity of the submersion.
Abstract: Akyol [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistics 2017; 462: 177-192] defined and studied conformal antiinvariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $\xi$ is a vertical vector field) from cosymplectic manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, and conformal antiinvariant submersions. More precisely, we mention many examples and obtain the geometries of the leaves of vertical distribution and horizontal distribution, including the integrability of the distributions, the geometry of foliations, some conditions related to total geodesicness, and harmonicity of the submersions. Finally, we consider a decomposition theorem on the total space of the new submersion.
11 citations
TL;DR: In this article, the authors studied the k-almost Ricci soliton and k-gradient Ricci s soliton on contact metric manifold and proved that if a compact k-contact metric is a k-approximation to a unit sphere S2n+1, then it is isometric to a sphere S 2 n+1.
Abstract: The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton then it is isometric to a unit sphere S2n+1. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci soliton where the potential vector field X is point wise collinear with the Reeb vector field {\xi} of the contact metric structure.
10 citations
TL;DR: Akyol M.A. as mentioned in this paper defined and studied conformal anti-invariant submersions from cosymplectic manifolds and obtained the geometries of the leaves of the inverted Reeb vector field.
Abstract: Akyol M.A. [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistic, 46(2), (2017), 177-192.] defined and studied conformal anti-invariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $\xi$ is a vertical vector field) from almost contact metric manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions and conformal anti-invariant submersions. More precisely, we mention lots of examples and obtain the geometries of the leaves of $\ker\pi_{*}$ and $(\ker\pi_{*})^\perp,$ including the integrability of the distributions, the geometry of foliations, some conditions related to totally geodesicness and harmonicty of the submersions. Finally, we consider a decomposition theorem on total space of the new submersion.
10 citations
DOI•
01 Jan 2018TL;DR: In this paper, the authors studied Ricci solitons in Kenmotsu manifolds under $D$-homothetic deformation and proved Ricci scalitons are shrinking in the case of potential vector fields orthogonal to Reeb vector fields and pointwise collinear with Reeb vectors.
Abstract: The aim of the present paper is to study Ricci solitons in Kenmotsu manifolds under $D$-homothetic deformation. We analyzed behaviour of Ricci solitons when potential vector field is orthogonal to Reeb vector field and pointwise collinear with Reeb vector field. Further we prove Ricci solitons in $D$-homothetically transformed Kenmotsu manifolds are shrinking.
8 citations
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TL;DR: In this paper, Chen-Cheng et al. showed that given a Sasaki structure with Reeb vector field and complex structure on its cone fixed, there exists a compact Sasaki manifold with transverse constant scalar curvature (cscs) if and only if the energy of the automorphism group is reduced proper modulo the identity component.
Abstract: We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of K\"ahler metrics with constant scalar curvature (csck) on compact K\"ahler manifolds Following their strategy, we prove that given a Sasaki structure (with Reeb vector field and complex structure on its cone fixed ), there exists a Sasaki structure with transverse constant scalar curvature (cscs) if and only if the $\mathcal{K}$-energy is reduced proper modulo the identity component of the automorphism group which preserves both the Reeb vector field and transverse complex structure Technically, the proof mainly consists of two parts The first part is a priori estimates for scalar curvature type equations which are parallel to Chen-Cheng's results in \cite{CC2, CC3} in Sasaki setting The second part is geometric pluripotential theory on a compact Sasaki manifold, building up on profound results in geometric pluripotential theory on K\"ahler manifolds There are notable, and indeed subtle differences in Sasaki setting (compared with K\"ahler setting) for both parts (PDE and pluripotential theory) The PDE part is an adaption of deep work of Chen-Cheng \cite{CC1, CC2, CC3} to Sasaki setting with necessary modifications While the geometric pluripotential theory on a compact Sasaki manifold has new difficulties, compared with geometric pluripotential theory in K\"ahler setting which is very intricate We shall present the details of geometric pluripotential on Sasaki manifolds in a separate paper \cite{HL} (joint work with Jun Li)
7 citations
TL;DR: In this article, the Ricci solitons on an almost cosymplectic manifold were studied and it was shown that they do not exist on such a manifold and that the potential vector field is the Reeb vector field.
Abstract: In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(\kappa,\mu)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimensional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $\eta$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton..
7 citations
13 Feb 2018
TL;DR: In this paper, the authors classify three-dimensional paracontact metric manifold whose Ricci operator Q is invariant along Reeb vector field, that is, $${\mathcal {L}} _{\xi }Q=0$$
Abstract: We classify three-dimensional paracontact metric manifold whose Ricci operator Q is invariant along Reeb vector field, that is, $${\mathcal {L}} _{\xi }Q=0$$
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2 citations
TL;DR: In this paper, a new family of non-degenerate Hopf real hypersurfaces whose shape operator is diagonalisable and one Hopf example with degenerate metric and non-diagonalisability shape operator was constructed.
Abstract: We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations for the almost contact metric structure. To make the theory meaningful, we construct new families of non-degenerate Hopf real hypersurfaces whose shape operator is diagonalisable, and one Hopf example with degenerate metric and non-diagonalisable shape operator. Next, we obtain a rigidity result. We classify those real hypersurfaces which are $\eta$-umbilical. As a consequence, we characterize some of our new examples as those whose Reeb vector field $\xi$ is Killing.
TL;DR: In this paper, the authors study three-dimensional homogeneous paracontact metric manifolds for which the Reeb vector field of the underlying parAContact structure satisfies a nullity condition.
Abstract: In this paper, we study three-dimensional homogeneous paracontact metric manifolds for which the Reeb vector field of the underlying paracontact structure satisfies a nullity condition. We give example of paraSasakian and non-paraSasakian (κ,μ)-manifolds. Finally, we exhibit explicit example of η-Einstein manifolds.
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TL;DR: In this paper, it was shown that there do not exist Ricci solitons on an almost cosymplectic manifold admitting a Ricci-soliton whose potential vector field is the Reeb vector field.
Abstract: In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(\kappa,\mu)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimesional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $\eta$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton.