scholarly journals A characterization of harmonic foliations by the volume preserving property of the normal geodesic flow

2002 ◽  
Vol 29 (10) ◽  
pp. 573-577
Author(s):  
Hobum Kim
Keyword(s):  
Geodesic Flow ◽  
Riemannian Foliation ◽  
Volume Form ◽  
Normal Connection ◽  
Riemannian Volume ◽  
Canonical Metric ◽  
Normal Geodesic ◽  
Harmonic Foliations ◽  
Volume Preserving

We prove that a Riemannian foliation with the flat normal connection on a Riemannian manifold is harmonic if and only if the geodesic flow on the normal bundle preserves the Riemannian volume form of the canonical metric defined by the adapted connection.

scholarly journals Vacuum structure and gravitational bags produced by metric-independent space–time volume-form dynamics

2015 ◽  
Vol 30 (22) ◽  
pp. 1550133 ◽  
Author(s):  
Eduardo Guendelman ◽  
Emil Nissimov ◽  
Svetlana Pacheva
Keyword(s):  
Gauge Field ◽  
Scalar Potential ◽  
Space Time ◽  
Kinetic Term ◽  
Volume Form ◽  
De Sitter ◽  
Field System ◽  
Riemannian Volume ◽  
A Charge ◽  
Nonlinear Gauge

We propose a new class of gravity-matter theories, describing [Formula: see text] gravity interacting with a nonstandard nonlinear gauge field system and a scalar “dilaton,” formulated in terms of two different non-Riemannian volume-forms (generally covariant integration measure densities) on the underlying space–time manifold, which are independent of the Riemannian metric. The nonlinear gauge field system contains a square-root [Formula: see text] of the standard Maxwell Lagrangian which is known to describe charge confinement in flat space–time. The initial new gravity-matter model is invariant under global Weyl-scale symmetry which undergoes a spontaneous breakdown upon integration of the non-Riemannian volume-form degrees of freedom. In the physical Einstein frame we obtain an effective matter-gauge-field Lagrangian of “k-essence” type with quadratic dependence on the scalar “dilaton” field kinetic term [Formula: see text], with a remarkable effective scalar potential possessing two infinitely large flat regions as well as with nontrivial effective gauge coupling constants running with the “dilaton” [Formula: see text]. Corresponding to each of the two flat regions we find “vacuum” configurations of the following types: (i) [Formula: see text] and a nonzero gauge field vacuum [Formula: see text], which corresponds to a charge confining phase; (ii) [Formula: see text] (“kinetic vacuum”) and ordinary gauge field vacuum [Formula: see text] which supports confinement-free charge dynamics. In one of the flat regions of the effective scalar potential we also find: (iii) [Formula: see text] (“kinetic vacuum”) and a nonzero gauge field vacuum [Formula: see text], which again corresponds to a charge confining phase. In all three cases, the space–time metric is de Sitter or Schwarzschild–de Sitter. Both “kinetic vacuums” (ii) and (iii) can exist only within a finite-volume space region below a de Sitter horizon. Extension to the whole space requires matching the latter with the exterior region with a nonstandard Reissner–Nordström–de Sitter geometry carrying an additional constant radial background electric field. As a result, we obtain two classes of gravitational bag-like configurations with properties, which on one hand partially parallel some of the properties of the solitonic “constituent quark” model and, on the other hand, partially mimic some of the properties of MIT bags in QCD phenomenology.

scholarly journals QUASI-MORPHISMS AND Lp-METRICS ON GROUPS OF VOLUME-PRESERVING DIFFEOMORPHISMS

2012 ◽  
Vol 04 (02) ◽  
pp. 255-270 ◽  
Author(s):  
MICHAEL BRANDENBURSKY
Keyword(s):  
Fundamental Group ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Volume Form ◽  
Oriented Manifold ◽  
Identity Component ◽  
Finite Dimensional ◽  
Lipschitz Embeddings ◽  
Volume Preserving

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form μ. We show that every homogeneous quasi-morphism on the identity component Diff 0(M, μ) of the group of volume-preserving diffeomorphisms of M, which is induced by a quasi-morphism on the fundamental group π1(M), is Lipschitz with respect to the Lp-metric on Diff 0(M, μ). As a consequence, assuming certain conditions on π1(M), we construct bi-Lipschitz embeddings of finite dimensional vector spaces into Diff 0(M, μ).

Méthode de martingales et flot géodésique sur une surface de courbure constante négative

2001 ◽  
Vol 21 (2) ◽  
pp. 421-441 ◽  
Author(s):  
J.-P. CONZE ◽  
S. LE BORGNE
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Discrete Time ◽  
Geodesic Flow ◽  
Hölder Functions ◽  
The Central Limit Theorem ◽  
Negative Constant ◽  
Nous Montrons ◽  
Unit Tangent Bundle

Let ({\cal T}^1 S, m, (T^t){t \in \mathbb{R}}) be the geodesic flow on the unit tangent bundle of a surface S of negative constant curvature and finite volume. We show that every Hölder function on {\cal T}^1 S is, for the discrete time action of the geodesic flow, homologous to a martingale increment. From this representation follow the central limit theorem and its improvements, and a characterization of Hölder functions which are coboundaries in the class of measurable functions.Soit ({\cal T}^1 S, m, (T^t)_{t \in \mathbb{R}}) le flot géodésique sur le fibré unitaire d'une surface S de courbure négative constante de volume fini. Nous montrons que toute fonction höldérienne sur {\cal T}^1 S est, pour l'action du flot géodésique à temps discret, homologue à un accroissement de martingale. Cette représentation permet d'obtenir le théorème de la limite centrale et ses extensions, et de caractériser les fonctions höldériennes qui sont des cobords dans la classe des fonctions mesurables.

scholarly journals Cosmological aspects of a unified dark energy and dust dark matter model

2016 ◽  
Vol 32 (01) ◽  
pp. 1750006 ◽  
Author(s):  
Denitsa Staicova ◽  
Michail Stoilov
Keyword(s):  
Phase Transition ◽  
Dark Matter ◽  
Dark Energy ◽  
Maximal Rank ◽  
Volume Form ◽  
Riemannian Volume ◽  
Type Ia ◽  
Lower Energy Density ◽  
Supernovae Type Ia ◽  
Universe Evolution

Recently, a model of modified gravity plus single scalar field was proposed, in which the scalar couples both to the standard Riemannian volume form given by the square root of the determinant of the Riemannian metric, as well as to another non-Riemannian volume form given in terms of an auxiliary maximal rank antisymmetric tensor gauge field. This model provides an exact unified description of both dark energy (via dynamically generated cosmological constant) and dark matter (as a “dust” fluid due to a hidden nonlinear Noether symmetry). In this paper, we test the model against Supernovae type Ia experimental data and investigate the future Universe evolution which follows from it. Our results show that this model has very interesting features allowing various scenarios of Universe evolution and in the same time perfectly fits contemporary observational data. It can describe exponentially expanding or finite expanding Universe and moreover, a Universe with phase transition of first kind. The phase transition occurs to a new, emerging at some time ground state with lower energy density, which affects significantly the Universe evolution.

Calibrations and laminations

2016 ◽  
Vol 162 (1) ◽  
pp. 151-171
Author(s):  
VICTOR BANGERT ◽  
XIAOJUN CUI
Keyword(s):  
Riemannian Manifold ◽  
Cohomology Class ◽  
Compact Manifold ◽  
Volume Form ◽  
Riemannian Volume

AbstractA calibration of degree k ∈ ℕ on a Riemannian manifold M is a closed differential k-form θ such that the integral of θ over every k-dimensional, oriented submanifold N is smaller or equal to the Riemannian volume of N. A calibration θ is said to calibrate N if θ restricts to the oriented volume form of N. We investigate conditions on a calibration θ that ensure the existence of submanifolds calibrated by θ. The cases k = 1 and k > 1 turn out to be essentially different. Our main result says that, on a compact manifold M, a calibration θ calibrates a lamination if θ is simple, of class C1, and if θ has minimal comass norm in its cohomology class.

scholarly journals Dark energy and dark matter from hidden symmetry of gravity model with a non-Riemannian volume form

2015 ◽  
Vol 75 (10) ◽  
Author(s):  
Eduardo Guendelman ◽  
Emil Nissimov ◽  
Svetlana Pacheva
Keyword(s):  
Dark Matter ◽  
Dark Energy ◽  
Gravity Model ◽  
Volume Form ◽  
Hidden Symmetry ◽  
Riemannian Volume

HOMOGENEOUS SPACES OF NONPOSITIVE CURVATURE AND THEIR GEODESIC FLOW

1995 ◽  
Vol 06 (02) ◽  
pp. 279-296 ◽  
Author(s):  
JENS HEBER
Keyword(s):  
Homogeneous Space ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Homogeneous Spaces ◽  
Nonpositive Curvature ◽  
Partial Classification ◽  
Higher Rank ◽  
Unit Tangent Bundle ◽  
Volume Preserving

Consider the geodesic flow on the unit tangent bundle SH of a 1-connected, irreducible homogeneous space H of nonpositive curvature. We prove that any flow invariant, isometry invariant C0-function on SH is necessarily constant, unless H is symmetric of higher rank. As the main applications, we obtain rigidity and partial classification results for spaces H whose geodesic symmetries are (asymptotically) volume-preserving.

scholarly journals Characterizations of Trivial Ricci Solitons

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Hana Alsodais
Keyword(s):  
Scalar Curvature ◽  
Potential Field ◽  
Geodesic Flow ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Constant Length

Finding characterizations of trivial solitons is an important problem in geometry of Ricci solitons. In this paper, we find several characterizations of a trivial Ricci soliton. First, on a complete shrinking Ricci soliton, we show that the scalar curvature satisfying a certain inequality gives a characterization of a trivial Ricci soliton. Then, it is shown that the potential field having geodesic flow and length of potential field satisfying certain inequality gives another characterization of a trivial Ricci soliton. Finally, we show that the potential field of constant length satisfying an inequality gives a characterization of a trivial Ricci soliton.

scholarly journals A characterization of 3D steady Euler flows using commuting zero-flux homologies

2020 ◽  
pp. 1-16
Author(s):  
DANIEL PERALTA-SALAS ◽  
ANA RECHTMAN ◽  
FRANCISCO TORRES DE LIZAUR
Keyword(s):  
Euler Equations ◽  
Vector Fields ◽  
Higher Dimensions ◽  
Riemannian Metric ◽  
Euler Flows ◽  
Homological Characterization ◽  
Steady Solutions ◽  
Volume Preserving

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.

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