Flat Manifolds Research Articles

The Gauss-Bonnet-Chern center of mass for asymptotically flat manifolds

The Gauss-Bonnet-Chern center of mass for asymptotically flat manifolds

Mechanics of geodesics in information geometry and black hole thermodynamics

In this paper, we shall discuss the theory of geodesics in information geometry, and an application in astrophysics. We will study how gradient flows in information geometry describe geodesics, explore the related mechanics by introducing a constraint, and apply our theory to Gaussian model and black hole thermodynamics. Thus, we demonstrate how deformation of gradient flows leads to more general Randers–Finsler metrics, describe Hamiltonian mechanics that derive from a constraint, and prove duality via canonical transformation. We also verified our theories for a deformation of the Gaussian model, and described dynamical evolution of flat metrics for Kerr and Reissner–Nordström black holes.

Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds

Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds

Some results on four-manifolds with nonnegative biorthogonal curvature

Let (M,g) be a 4-dimensional compact oriented Riemannian manifold with nonnegative biorthogonal curvature and W− be the anti-self-dual component of the Weyl curvature tensor W. If M has constant scalar curvature and |W−| of M is constant, or if M has harmonic Weyl tensor and detW− or |W−| of M is constant, then we give two classification theorems for M. As two applications, a 4-dimensional compact oriented Einstein manifold with nonnegative sectional curvature whose detW− or |W−| is constant is isometric to one of S4, CP2, S2×S2 or a flat manifold; a 4-dimensional compact oriented self-dual Riemannian manifold with positive sectional curvature and constant scalar curvature either is conformal to S4, CP2, or is diffeomorphic to CP2♯CP2, CP2♯CP2♯CP2.

On the Pontrjagin classes of spray manifolds

Abstract The characterization of projectively flat Finsler metrics on an open subset in $R^n$ is the Hilbert’s fourth problem in the regular case. Locally projectively flat Finsler manifolds form an important class of Finsler manifolds. Every Finsler metric induces a spray on the manifold via geodesics. Therefore, it is a natural problem to investigate the geometric and topological properties of manifolds equipped with a spray. In this paper, we study the Pontrjagin classes of a manifold equipped with a locally projectively flat spray and show that such manifold must have zero Pontrjagin classes.

Manifolds with harmonic Weyl tensor and nonnegative curvature operator of the second kind

We show that a compact manifold with harmonic Weyl tensor and nonnegative curvature operator of the second kind is globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. Moreover, we obtain that a compact manifold with harmonic curvature and nonnegative curvature operator of the second kind is isometric to a space of constant curvature. These improve Kashiwada's result that Riemannian manifolds with harmonic curvature tensor for which the curvature operator of the second kind is nonnegative and positive are locally symmetric spaces and constant curvature spaces, respectively.

Intrinsic flat stability of the positive mass theorem for asymptotically hyperbolic graphical manifolds

The rigidity of the Riemannian positive mass theorem for asymptotically hyperbolic manifolds states that the total mass of such a manifold is zero if and only if the manifold is isometric to the hyperbolic space. This leads to study the stability of this statement, that is, if the total mass of an asymptotically hyperbolic manifold is almost zero, is this manifold close to the hyperbolic space in any way? Motivated by the work of Huang, Lee and Sormani for asymptotically flat graphical manifolds with respect to intrinsic flat distance, we show the intrinsic flat stability of the positive mass theorem for a class of asymptotically hyperbolic graphical manifolds by adapting the positive answer to this question provided by Huang, Lee and the third named author.

Hořava–Lifshitz F(R) Theories and the Swampland

The compatibility between the de Sitter Swampland conjecture and Hořava–Lifshitz F(R¯) theories with a flat FLRW metric is studied. We first study the standard f(R) theories and show that the only way in which the dS conjecture can be made independent of R is by considering a power law of the form f(R)∼Rγ. The conjecture and the consistency of the theory puts restrictions on γ to be greater but close to one. For F(R¯) theories described by its two parameters λ and μ, we use the equations of motion to construct the function starting with an ansatz for the scale factor in the Jordan frame of the power law form. By performing a conformal transformation on the three metric to the Einstein frame, we can obtain an action of gravity plus a scalar field by relating the parameters of the theory. The non-projectable and projectable cases are studied and the differences are outlined. The obtained F(R¯) function consists of terms of the form R¯γ with the possibility of having negative power terms. The dS conjecture leads to inequalities for the λ parameter; in both versions, it becomes restricted to be greater but close to 1/3. We can also study the general case in which μ and λ are considered as independent. The obtained F function has the same form as before. The consistency of the theory and the dS conjecture lead to a set of inequalities on both parameters that are studied numerically. In all cases, λ is restricted by μ around 1/3, and we obtain λ→1/3 if μ→0. We consider the f(R) limit μ,λ→1 and we obtain consistent results. Finally, we study the case of a constant Hubble parameter. The dS conjecture can be fulfilled by restricting the parameters of the theory; however, the constraint makes this compatibility exclusive to these kinds of theories.

On isoperimetric problem in 2-dimensional Randers space

In this paper, we prove that the circle centered at the origin in B2(δξ) is a proper maximum of the isoperimetric problem in a 2-dimensional Randers space endowed with 3-parameter family of non-locally projectively flat Finsler metrics of non-constant isotropic S-curvature.

Gravitational lensing effect in the Universe with a Finsler background

The gravitational deflection of light is a common effect for the massive objects in galactic astronomy. Recently, several universal models with Finsler background were introduced to solve the abnormal phenomena. In this manuscript, we consider the deflection of gravitational lens in a general Finsler space. Firstly, with the assistance of the Landsberg angle, we introduce four definitions of the deflection angle, which are not equal mutually in general case. Secondly, we introduce an important concept called asymptotically symmetric Minkowskian flat (ASMF) manifold, on which we define the global normal deflection angle (GNDA) as well as the global tangent deflection angle (GTDA). Lastly, we present an integral expression of the GNDA on an ASMF manifold. This result suggests to us that the non-Riemannian Finsler quantities may contain the information of dark matter.

The Yamabe flow on asymptotically flat manifolds

Abstract We study the Yamabe flow starting from an asymptotically flat manifold ( M n , g 0 ) (M^{n},g_{0}) . We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if Y ⁢ ( M , [ g 0 ] ) > 0 Y(M,[g_{0}])>0 , and show that the flow does not converge otherwise. If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow.

Global Stability for Charged Scalar Fields in an Asymptotically Flat Metric in Harmonic Gauge

We prove global stability for the charged scalar field system on a background spacetime, which is close to 1+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1+3$$\\end{document}-dimensional Minkowski space and whose outward light cones converge to those for the Schwarzs-child metric at null infinity. The key technique to this proof is the use of a modified null frame, depending only on the mass M of the metric, which captures the asymptotic behavior of the metric at future null infinity. Our results are analogous to results obtained in Minkowski space by Lindblad and Sterbenz in (IMRP Int Math Res Pap 2006:52976, 2006) up to a change in coordinates, and will in the sequel be used to prove the full structure of the Einstein–charge scalar field system in these modified harmonic coordinates.

Locally conformally flat affine hyperspheres with parallel Ricci tensor

Trying to extend the result of Vrancken et al. (1991) [20] which established the complete classification of locally strongly convex affine hyperspheres with constant sectional curvature, we consider the classification problem of n-dimensional locally strongly convex affine hyperspheres with locally conformally flat affine metrics. As the main result of this paper, we solved the problem under the additional condition that the affine metric has parallel Ricci tensor with respect to its Levi-Civita connection.

BRST BMS4 symmetry and its cocycles from horizontality conditions

The BRST structure of the extended Bondi-Metzner-Sachs symmetry group of asymptotically flat manifolds is investigated using the recently introduced framework of the Beltrami field parametrization of four-dimensional metrics. The latter identifies geometrically the two physical degrees of freedom of the graviton as fundamental fields. The graded BRST BMS4 nilpotent differential operator relies on four horizontality conditions giving a Lagrangian reformulation of the asymptotic BMS4 symmetry. A series of cocycles is found which indicate the possibility of anomalies for three-dimensional Lagrangian theories to be built in the null boundaries of asymptotically flat spaces from the principle of BRST BMS4 invariance.

Mass, center of mass and isoperimetry in asymptotically flat 3-manifolds

Mass, center of mass and isoperimetry in asymptotically flat 3-manifolds

Geodesics on metrics of Eguchi–Hanson type

Geodesic equations are solved when at least two of theta , phi and psi are constant, or r is constant, on scalar flat metrics of Eguchi–Hanson type. They can also be solved also on Eguchi–Hanson metrics which are Ricci flat if only phi is constant. However, the explicit solution of the geodesic equations is not available yet if only psi is constant.

Compacitification and Positive Mass Theorem for Fibered Euclidean End

We consider the Positive Mass Theorem for Riemannian manifolds (M^{n},g) with the asymptotic end (mathbb {R}^{k}times X^{n-k}, g_{mathbb {R}^{k}}+g_{X}) (kge 3) by studying the corresponding compactification problem. Here (X, g_X) is a compact scalar flat manifold. We show that the Positive Mass Theorem holds if certain generalized connected sum admits no metric of positive scalar curvature. Moreover we establish the rigidity result, namely, the mass is zero iff M is isometric to mathbb {R}^{k}times X^{n-k}.

A MacDowell–Mansouri-type formulation for conformally flat Einstein manifolds

We study a pure connection formulation plus algebraic constraints in four spacetime dimensions where the gauge group [Formula: see text]. We show that the action has, as particular cases, the MacDowell–Mansouri and the Stelle–West formulations for gravity. Also, under adequate specification of the constraint terms, we obtain a subset of Einstein manifolds that are torsionless conformally flat manifolds.

A constrained cosmological model in [formula omitted] gravity

In this article, we study the expanding nature of the universe in the context of f(R,Lm) gravity theory, here R represents the Ricci scalar and Lm is the matter Lagrangian density. With a specific form of f(R,Lm), we obtain the field equations for flat FLRW metric. We parametrize the deceleration parameter in terms of the Hubble parameter and from here we find four free parameters, which are constraints and estimated by using H(z), Pantheon, and their joint data sets. Further, we investigate the evolution of the deceleration parameter which depicts a transition from the deceleration to acceleration phases of the universe. The evolutionary behaviour of energy density, pressure, and EoS parameter shows that the present model is an accelerated quintessence dark energy model. To compare our model with the ΛCDM model we use some of the diagnostic techniques. Thus, we find that our model in f(R,Lm) gravity supports the recent standard observational studies and delineates the late-time cosmic acceleration.

Shadows of rotating black holes in plasma environments with aberration effects

The shadows of black holes encode significant information about the properties of black holes and the spacetime surrounding them. So far, the effects of dispersive media, such as plasma, and relativistic aberration on the propagation of light around compact objects have been treated separately in the literature. In this paper, we will employ the Konoplya, Stuchl\'{\i}k, and Zhidenko family of stationary, axially symmetric, and asymptotically flat metrics to describe the spacetime around rotating black holes. We will study how the parameters of the black hole, the chromatic effects resulting from the presence of a nonmagnetized, pressureless plasma environment, and the effects of relativistic aberration of a moving observer modify the morphology of the shadow.