Flat Manifolds Research Articles

Adiabatic limits of anti-self-dual connections on collapsed $K3$ surfaces

We prove a convergence result for a family of Yang–Mills connections over an elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is projective, admits a section, and has singular fibers of Kodaira type $I_1$ and type $II$. Let $\Xi_{t_k}$ be a sequence of $SU(n)$ connections on a principal $SU(n)$ bundle over $M$, that are anti-self-dual with respect to a sequence of Ricci flat metrics collapsing the fibers of $M$. Given certain non-degeneracy assumptions on the spectral covers induced by $\overline{\partial}_{\Xi_{t_k}}$, we show that away from a finite number of fibers, the curvature $F_{\Xi_{t_k}}$ is locally bounded in $C^0$, the connections converge along a subsequence (and modulo unitary gauge change) in $L^p_1$ to a limiting $L^p_1$ connection $\Xi_0$, and the restriction of $\Xi_0$ to any fiber is $C^{1,\alpha}$ gauge equivalent to a flat connection with holomorphic structure determined by the sequence of spectral covers. Additionally, we relate the connections $\Xi_{t_k}$ to a converging family of special Lagrangian multi-sections in the mirror HyperKahler structure, addressing a conjecture of Fukaya in this setting.

Limits of Blaschke metrics

We find a compactification of the SL(3,R)-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.

On the null-timelike boundary for Maxwell and spin-2 fields in asymptotically flat spaces

We study the Maxwell equation and the spin-2 field equation in Bondi–Sachs coordinates associated with an asymptotically flat Lorentzian metric. We consider the mixed boundary/initial value problem, where the initial data are imposed on a null hypersurface and a boundary value is prescribed on a timelike hypersurface. We establish Sobolev [Formula: see text] space-time estimates for solutions and their asymptotic expansions at the null infinity.

Exact solution of the fiels equation of the generalized cosmological F(R,T) model

We considered cosmological model within the framework of the generalized F(R,T) gravityin flat FLRW metric.  Earlier, an exponential solution was shown for the generalized case, which describes the inflationary stage of the evolution of the Universe. In this paper, it is shown that for the particular case of the Starobinsky model, the generalized model has, amongother things, a power-law solution that describes the dust-like stage  and  the  stage  of  radiation  dominance.It  should  be  noted  that  this  solution  was  obtained  by  the  analytical  method  as  a  solution  to  the  Euler-Lagrange  equation  for  this  model,  taking  into  account  the  functions  u,  v–  describing  the  relationship  between  the  curvature  scalar  and  the  torsion  scalar,  and  depending not only on the first, but also on the second time derivative of the scale factor. This solution is of interest, since it shows not only that this model has a solution, but also that this solution can describe the modern observable Universe for more complex forms u, v than described earlier.This solution allows other related cosmological parameters to be obtained accordingly. The paper presents the parameter for the equation of state, which describes the stage of the evolution of the Universe. Keywords:generalized cosmological model, scale factor, Starobinsky model

On Projectively Flat Finsler Warped Product Metrics of Constant Flag Curvature

In this paper, we study locally projectively flat Finsler metrics of constant flag curvature. We find equations that characterize these metrics by warped product. Using the obtained equations, we manufacture new locally projectively flat Finsler warped product metrics of vanishing flag curvature. These metrics contain the metric introduced by Berwald and the spherically symmetric metric given by Mo-Zhu.

Profinite genus of the fundamental groups of compact flat manifolds with holonomy group of prime order

Abstract In this article, we calculate the profinite genus of the fundamental group of an 𝑛-dimensional compact flat manifold 𝑋 with holonomy group of prime order. As consequence, we prove that if n ⩽ 21 n\leqslant 21 , then 𝑋 is determined among all 𝑛-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of the profinite completion of the fundamental group of 𝑋 in terms of the representation genus of its holonomy group.

Twisted bilayer graphene. VI. An exact diagonalization study at nonzero integer filling

Using exact diagonalization, we study the projected Hamiltonian with Coulomb interaction in the 8 flat bands of first magic angle twisted bilayer graphene. Employing the U(4) (U(4)$\times$U(4)) symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent which allows for study around $\nu=\pm 3,\pm2,\pm1$ fillings. In the first chiral limit $w_0/w_1=0$ where $w_0$ ($w_1$) is the $AA$ ($AB$) stacking hopping, we find that the ground-states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge $\pm1$ excitations found in [arXiv:2009.14200] are the lowest charge excitations up to system sizes $8\times8$ (for restricted Hilbert space) in the chiral-flat limit. We also find that the Flat Metric Condition (FMC) used in [arXiv:2009.11301,2009.11872,2009.12376,2009.13530,2009.14200] for obtaining a series of exact ground-states and excitations holds in a large parameter space. For $\nu=-3$, the ground state is the spin and valley polarized Chern insulator with $\nu_C=\pm1$ at $w_0/w_1\lesssim0.9$ (0.3) with (without) FMC. At $\nu=-2$, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when $w_0/w_1\gtrsim0.5t$ where $t\in[0,1]$ is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [arXiv:2009.13530]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [arXiv:2009.13530]. For $\nu=-3$ with/without FMC, when $w_0/w_1$ is large, the finite-size gap $\Delta$ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at $\nu=-3$ suggests a competition among (nematic) metal, momentum $M_M$ ($\pi$) stripe and $K_M$-CDW orders at large $w_0/w_1$.

Concircularity on GRW-space-times and conformally flat spaces

There are two smooth functions [Formula: see text] and [Formula: see text] associated to a nontrivial concircular vector field [Formula: see text] on a connected Riemannian manifold [Formula: see text], called potential function and connecting function. In this paper, we show that presence of a timelike nontrivial concircular vector field influences the geometry of generalized Robertson–Walker space-times. We use a timelike concircular vector field [Formula: see text] on an [Formula: see text] -dimensional connected conformally flat Lorentzian manifold, [Formula: see text], to find a characterization of generalized Robertson–Walker space-time with fibers Einstein manifolds. It is interesting to note that for [Formula: see text] the concircular vector field annihilates energy-momentum tensor and also that in this case the potential function [Formula: see text] is harmonic. In the second part of this paper, we show that presence of a nontrivial concircular vector field [Formula: see text] with connecting function [Formula: see text] on a complete and connected [Formula: see text] -dimensional conformally flat Riemannian manifold [Formula: see text], [Formula: see text], with Ricci curvature [Formula: see text] non-negative, satisfying [Formula: see text], is necessary and sufficient for [Formula: see text] to be isometric to either a sphere [Formula: see text] or to the Euclidean space [Formula: see text], where [Formula: see text] is the scalar curvature.

Quantum phase space description of a cosmological minimal massive bigravity model

Bimetric gravity theories describes gravitational interactions in the presence of an extra spin-2 field. The Hassan-Rosen (HR) nonlinear massive minimal bigravity theory is a ghost-free bimetric theory formulated with respect a flat, dynamical reference metric. In this work the deformation quantization formalism is applied to a HR cosmological model in the minisuperspace. The quantization procedure is performed explicitly for quantum cosmology in the minisuperspace. The Friedmann-Lema\^itre-Robertson-Walker (FLRW) model with flat metrics is worked out and the computation of the Wigner functions for theHartle-Hawking, Vilenkin and Linde wavefunctions are done numerically and, in the Hartle-Hawking case, also analytically. From the stability analysis in the quantum minisuper phase space it is found an interpretation of the mass of graviton as an emergent cosmological constant and as a measure of the deviation of classical behavior of the Wigner functions. Also, from the Hartle-Hawking case, an interesting relation between the curvature and the mass of graviton in a cusp catastrophe surface is discussed

L^{p} harmonic 1-forms on conformally flat Riemannian manifolds

In this paper, we establish a finiteness theorem for L^{p} harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on L^{2} harmonic 1-forms.

∗-Conformal η-Ricci soliton on Sasakian manifold

In this paper, we study ∗-Conformal [Formula: see text]-Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting ∗-Conformal [Formula: see text]-Ricci soliton. We obtain some significant results on ∗-Conformal [Formula: see text]-Ricci soliton in Sasakian manifolds satisfying [Formula: see text], [Formula: see text], [Formula: see text] [Formula: see text], where [Formula: see text] is Pseudo-projective curvature tensor. The conditions for ∗-Conformal [Formula: see text]-Ricci soliton on [Formula: see text]-conharmonically flat and [Formula: see text]-projectively flat Sasakian manifolds have been obtained in this paper. Lastly we give an example of five-dimensional Sasakian manifolds satisfying ∗-Conformal [Formula: see text]-Ricci soliton.

Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds

In this article, we discuss the global aspects of the geometry of locally conformally flat (complete and compact) Riemannian manifolds. In particular, the article reviews and improves some results (e.g., the conditions of compactness and degeneration into spherical or flat space forms) on the geometry “in the large" of locally conformally flat Riemannian manifolds. The results presented here were obtained using the generalized and classical Bochner technique, as well as the Ricci flow.

Approximate analytical description of apparent horizons for initial data with momentum and spin

We construct analytical initial data for a slowly moving and rotating black hole for generic orientations of the linear momentum and the spin. We solve the Hamiltonian constraint approximately and work out the properties of the apparent horizon and show the dependence of its shape on the angle between the spin and the linear momentum. In particular a dimple, whose location depends on the mentioned angle, arises on the 2-sphere geometry of the apparent horizon. We exclusively work in the case of conformally flat initial metrics.

Cylinder curves in finite holonomy flat metrics

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order [Formula: see text], the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when [Formula: see text] is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core curves form an infinite diameter subset of the curve complex. In this paper we focus on the case [Formula: see text] and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if [Formula: see text] and the surface is fully punctured, then the embedded cylinder curves form a finite diameter subset of the curve complex. The same analysis shows that the embedded cylinder curves can only have infinite diameter when the metric has a very specific form. Using this we characterize precisely when the embedded cylinder curves accumulate on a point in the Gromov boundary.

The classification of flat Riemannian metrics on the plane

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of naturally occurring, non-isometric metrics that are relevant and useful. Remarkably, the study and classification of all flat Riemannian metrics on the plane—as a subject—is new to the literature. Much of our research focuses on conformal metrics of the form \(e^{2\varphi }g_0\), where \(\varphi : {\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a harmonic function and \(g_0\) is the standard Euclidean metric on \({\mathbb {R}}^2\). We find that all such metrics, which we call “harmonic,” arise from Riemann surfaces.

The Farthest Point Map on the Regular Octahedron

In this paper, we give a complete description of the farthest point map, as defined on the regular octahedron with its intrinsic flat cone metric. As a consequence we show that every well-defined orbit of the map converges to the 1-skeleton of the barycentric subdivision of the triangulation.

Transformation groups of certain flat affine manifolds

In this paper we characterize the group of affine transformations of a flat affine simply connected manifold whose developing map is a diffeomorphism. This is proved by making use of some simple facts about homeomorphisms of $\mathbb{R}^n$ preserving open connected sets. We show some examples where the characterization is useful.

Stringy tachyonic instabilities of non-supersymmetric Ricci flat backgrounds

Superstring/M-theory compactified on compact Ricci flat manifolds have recently been conjectured to exhibit instabilities whenever the metrics do not have special holonomy. We use worldsheet conformal field theory to investigate instabilities of Type II superstring theories on compact, Ricci flat, spin 3-manifolds including a worldsheet description of their spin structures. The instabilities are signalled by the appearance of stringy tachyons at small radius and a negative (1-loop) vacuum energy density at large radius. We briefly discuss the extension to higher dimensions.

Similarity Classes of Tetrahedra

We study the topology and geometry of the space $$\varUpsilon $$ of shapes of abstract tetrahedra, i.e. flat metrics on the 2-sphere with four conical singularities, up to equivalence by orientation-preserving similarities. There is no labeling of the conical singularities; two of them with the same cone-angle can be interchanged by a similarity. We are dealing with the geometric structure introduced by Thurston through the area viewed as a quadratic form. $$\varUpsilon $$ is shown to be a quotient of a real analytic fibration over the 3-dimensional space of cone-angles, with each fiber being a thrice-punctured 2-sphere of constant curvature. In particular, $$\varUpsilon $$ is shown to be a contractible orbifold of dimension five. It is also observed that the metric given by the area form degenerates transversally to the fibers. A description of the real analytic structure of $$\varUpsilon $$ is given by hypergeometric functions.

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.