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 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 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 Complete conformal metrics with prescribed scalar curvature on subdomains of a compact manifold

1993 ◽  
Vol 132 ◽  
pp. 155-173
Author(s):  
Shin Kato ◽  
Shin Nayatani
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Smooth Function ◽  
Compact Manifold ◽  
Second Order ◽  
Conformal Metrics ◽  
Derivatives Of

Let (M, g) be a Riemannian manifold of dimension n≥ 3 and ĝanother metric on M which is pointwise conformai to g. It can be written where u is a positive smooth function on M. Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equationwhere ▽u denotes the Laplacian of u, defined with respect to the metric g.

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.

scholarly journals Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Albert Fathi
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Jacobi Equation ◽  
Compact Manifold ◽  
Complete Riemannian Manifold ◽  
Time Dependent ◽  
Jacobi Equations ◽  
Uniformly Continuous ◽  
Hamilton Jacobi Equation

AbstractIf $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

scholarly journals Generalized Goldberg Formula

2016 ◽  
Vol 59 (3) ◽  
pp. 508-520
Author(s):  
Antonio De Nicola ◽  
Ivan Yudin
Keyword(s):  
Riemannian Manifold ◽  
Compact Manifold ◽  
Differential Forms ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Sasakian Manifolds ◽  
De Rham ◽  
Locally Conformally Kähler

AbstractIn this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed p-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel I. Goldberg for the case of 1-forms. As first examples of application we obtain new identities on locally conformally Kähler manifolds and quasi-Sasakian manifolds. Moreover, we prove that under suitable conditions a certain subalgebra of differential forms in a compact manifold is quasi-isomorphic as a CDGA to the full de Rham algebra.

scholarly journals FOUR-MANIFOLDS WITH POSITIVE CURVATURE

2020 ◽  
pp. 1-13
Author(s):  
R. DIÓGENES ◽  
E. RIBEIRO ◽  
E. RUFINO
Keyword(s):  
Riemannian Manifold ◽  
Compact Manifold ◽  
Positive Curvature ◽  
Conformally Flat ◽  
Complex Projective Plane ◽  
Connected Sum ◽  
Sectional Curvatures ◽  
Four Manifolds ◽  
Complex Projective ◽  
Flat Riemannian Manifold

Abstract In this note, we prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M 4 is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2}$ , provided that the sectional curvatures all lie in the interval $\left[ {{{3\sqrt {3 - 5} } \over 4}, 1} \right]$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.

scholarly journals PATH INTEGRALS ON A COMPACT MANIFOLD WITH NON-NEGATIVE CURVATURE

2007 ◽  
Vol 19 (09) ◽  
pp. 967-1044 ◽  
Author(s):  
ADRIAN P. C. LIM
Keyword(s):  
Compact Manifold ◽  
Compact Riemannian Manifold ◽  
Path Integrals ◽  
Energy Functional ◽  
Normalization Constant ◽  
Finite Dimensional ◽  
Riemannian Volume ◽  
Volume Measure ◽  
Constant E ◽  
Geodesic Paths

A typical path integral on a manifold, M is an informal expression of the form [Formula: see text] where H(M) is a Hilbert manifold of paths with energy E(σ) < ∞, f is a real-valued function on H(M), [Formula: see text] is a "Lebesgue measure" and Z is a normalization constant. For a compact Riemannian manifold M, we wish to interpret [Formula: see text] as a Riemannian "volume form" over H(M), equipped with its natural G1 metric. Given an equally spaced partition, [Formula: see text] of [0, τ], let [Formula: see text] be the finite dimensional Riemannian submanifold of H(M) consisting of piecewise geodesic paths adapted to [Formula: see text]. Under certain curvature restrictions on M, it is shown that [Formula: see text] where [Formula: see text] is a "normalization" constant, E : H(M) → [0,∞) is the energy functional, [Formula: see text] is the Riemannian volume measure on [Formula: see text], ν is Wiener measure on continuous paths in M, and ρ is a certain density determined by the curvature tensor of M.

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