Isometric Flow Research Articles

Cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures

We consider the existence of cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures on the following classes of torsion-free G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds: the Euclidean R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} with its standard G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structure, metric cylinders over Calabi–Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant–Salamon G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

Hard Lefschetz Property for Isometric Flows

The hard Lefschetz property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property), but a new version of the HLP has been recently given in terms of duality of the cohomology of the manifold itself in [1]. Both properties were proved to be equivalent (see [2]) in the case of K-contact flows. In this paper, we extend both versions of the HLP (transverse and not) to the more general category of isometric flows, and show that they are equivalent. We also give some explicit examples which illustrate the categories where the HLP could be considered.

On real hypersurfaces of 𝕊²×𝕊²

In this paper, regarding the Riemannian product S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 of two unit 2 2 -spheres as a Kähler surface, we study its real hypersurfaces with typical geometric properties. First, we classify the real hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with isometric Reeb flow and then, by using a Simons’ type inequality, a characterization of these compact real hypersurfaces is provided. Next, we classify Hopf hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with constant product angle function. Finally, we classify Hopf hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with parallel Ricci tensor.

Effect of nitrate supplementation on skeletal muscle motor unit activity during isometric blood flow restriction exercise

BackgroundNitrate (NO3−) supplementation has been reported to lower motor unit (MU) firing rate (MUFR) during dynamic resistance exercise; however, its impact on MU activity during isometric and ischemic exercise is unknown.PurposeTo assess the effect of NO3− supplementation on knee extensor MU activities during brief isometric contractions and a 3 min sustained contraction with blood flow restriction (BFR).MethodsSixteen healthy active young adults (six females) completed two trials in a randomized, double-blind, crossover design. Trials were preceded by 5 days of either NO3− (NIT) or placebo (PLA) supplementation. Intramuscular electromyography was used to determine the M. vastus lateralis MU potential (MUP) size, MUFR and near fibre (NF) jiggle (a measure of neuromuscular stability) during brief (20 s) isometric contractions at 25% maximal strength and throughout a 3 min sustained BFR isometric contraction.ResultsPlasma nitrite (NO2−) concentration was elevated after NIT compared to PLA (475 ± 93 vs. 198 ± 46 nmol L−1, p < 0.001). While changes in MUP area, NF jiggle and MUFR were similar between NIT and PLA trials (all p > 0.05), MUP duration was shorter with NIT compared to PLA during brief isometric contractions and the sustained ischemic contraction (p < 0.01). In addition, mean MUP duration, MUP area and NF jiggle increased, and MUFR decreased over the 3 min sustained BFR isometric contraction for both conditions (all p < 0.05).ConclusionsThese findings provide insight into the effect of NO3− supplementation on MUP properties and reveal faster MUP duration after short-term NO3− supplementation which may have positive implications for skeletal muscle contractile performance.

Bernoulli property for certain skew products over hyperbolic systems

We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider a hyperbolic map ( T , M , μ ) (T,M,\mu ) , where μ \mu is a Gibbs measure, an aperiodic Hölder continuous cocycle ϕ : M → R \phi :M\to \mathbb {R} with zero mean and a zero-entropy flow ( K t , N , ν ) (K_t,N,\nu ) . We then study the skew product T ϕ ( x , y ) = ( T x , K ϕ ( x ) y ) , \begin{equation*} T_\phi (x,y)=(Tx,K_{\phi (x)}y), \end{equation*} acting on ( M × N , μ × ν ) (M\times N,\mu \times \nu ) . We show that if ( K t ) (K_t) is of slow growth and has good equidistribution properties, then T ϕ T_\phi remains Bernoulli. In particular, our main result applies to ( K t ) (K_t) being a typical translation flow on a surface of genus ≥ 1 \geq 1 or a smooth reparametrization of isometric flows on T 2 \mathbb {T}^2 . This provides examples of non-algebraic, partially hyperbolic systems which are Bernoulli and for which the center is non-isometric (in fact might be weakly mixing).

Singular Riemannian flows and characteristic numbers

Let $M$ be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on $M$ to a small tubular neighborhood of each connected component of its singular stratum is foliated-diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes characteristic numbers of $M$ as the sum of residues associated to the infinitesimal foliation at the components of the singular stratum of the flow.

Group action on local dendrites

Let G be a group acting by homeomorphisms on a local dendrite X with countable set of endpoints. In this paper, it is shown that any minimal set M of G is either a finite orbit, or a Cantor set or a circle. Furthermore, we prove that if G is a finitely generated group, then the flow (G,X) is a pointwise recurrent flow if and only if one of the following two statements holds:(1)X=S1, and (G,S1) is a minimal flow conjugate to an isometric flow, or to a finite cover of a proximal flow;(2)(G,X) is pointwise periodic.

SMOOTHNESS OF ISOMETRIC FLOWS ON ORBIT SPACES AND APPLICATIONS

We prove here that given a proper isometric action K × M → M on a complete Riemannian manifold M, then every continuous isometric flow on the orbit space M/K is smooth, i.e., it is the projection of a K-equivariant smooth flow on the manifold M. As a direct corollary we infer the smoothness of isometric actions on orbit spaces. Another relevant application of our result concerns Molino’s conjecture, which states that the partition of a Riemannian manifold into the closures of the leaves of a singular Riemannian foliation is still a singular Riemannian foliation. We prove Molino’s conjecture for the main class of foliations considered in his book, namely orbit-like foliations.

Pointwise recurrent one-dimensional flows

Let (G, X) be a flow, where X is a graph and G is a finitely generated group. In this article, it is shown that (G, X) is a pointwise recurrent flow if and only if one of the following two statements holds: (1) X = 𝕊1, and (G, 𝕊1) is a minimal flow conjugate to an isometric flow, or to a finite cover of a proximal flow; (2) G is finite.

A Geometric study of Wasserstein spaces: Euclidean spaces

We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) "exotic" isometric flow. This contrasts with the case of higher-dimensional Euclidean spaces, where all isometries of the Wasserstein space preserve the shape of measures. We also study the curvature and various ranks of these spaces.

Regular and quasiregular isometric flows on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on smooth Riemannian manifolds. We describe the properties of the set of all points of finite (infinite) period for general isometric flows on Riemannian manifolds. It is shown that this flow is generated by an effective almost free isometric action of the group S1 if there are no points of infinite or zero period. In the last case, the set of periods is at most countable and generates naturally an invariant stratification with closed totally geodesic strata; the union of all regular orbits is an open connected dense subset of full measure.

Energy and Riemannian Flows

We define and compute the energy of 1-foliations on riemannian manifolds. We then derive the Euler-Lagrange equations associated with the energy. We also prove that Riemannian flows on manifolds of constant curvature are ritical if and only if they are isometric. Finally we prove that isometric flows on 3-manifolds are critical if and only if either they are transverse to 2-dimensional foliations or they provide K-contact structures.

On Harmonic Theory in Flows

AbstractRecently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of H-harmonic and H*-harmonic spaces associated to a Hörmander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

The Euler class for Riemannian flows

Let F be a Riemannian flow on a closed manifold M. The Gysin sequence relating the basic cohomology of F and the de Rham cohomology of M has been constructed for isometric flows in [6]. In this paper, we extend it to the case of Riemannian flows. We also give a geometric characterization of the vanishing of the Euler class similar to the one given in [9].

Isometric Stochastic Flows on Spheres

We consider isometric stochastic flows on the sphere Sn−1 with the same one point motion. In particular, we will show that when n>3, the set of such flows with Brownian motion as one point motion can be represented by a cube in some Euclidean space.

Invariants and flow geometry

We continue the study of Riemannian manifolds (M,g) equipped with an isometric flow $ℱ_ξ$ generated by a unit Killing vector field ξ. We derive some new results for normal and contact flows and use invariants with respect to the group of ξ-preserving isom

Toeplitz algebras associated to isometric flows

In this paper, we associate a Toeplitz C*-algebra T() to an isometric flow on M, and begin to study how the C*-algebraic properties of T((I)) are related to the geometric and topological properties of The algebra T() is defined as follows. Since each map qt is an isometry of M, it induces a unitary operator Ut on L2(M), whereM is endowed with the usual measure coming from the Riemannian metric. Let D be the infinitesimal generator of the group {Ut}, and let P be the positive spectral projection of D. Then T() is the C*-subalgebra of ,(L2(M)) generated by the set {PMfP: f C(M)}, where

Geodesic spheres and isometric flows

Geodesic spheres and isometric flows

Dirichlet polyhedra for cyclic groups in complex hyperbolic space

We prove that the Dirichlet fundamental polyhedron for a cyclic group generated by a unipotent or hyperbolic element γ \gamma acting on complex hyperbolic n n -space centered at an arbitrary point w w is bounded by the two hypersurfaces equidistant from the pairs w , γ w w,\gamma w and w , γ − 1 w w,{\gamma ^{ - 1}}w respectively. The proof relies on a convexity property of the distance to an isometric flow containing γ \gamma .

Processes Disjoint from Weak Mixing

We show that the family ${\mathcal {W}^ \bot }$ of ergodic measure preserving transformations which are disjoint from every weakly mixing m.p.t. properly contains the family $\mathcal {D}$ of distal ergodic m.p.t. In the topological case we show that $\mathcal {P}\mathcal {I}$, the family of proximally isometric flows is properly contained in the family $\mathcal {M}({\mathcal {W}^ \bot })$ of multipliers for ${\mathcal {W}^ \bot }$.