Flat Manifolds Research Articles

On conformally flat centroaffine hypersurfaces with semi-parallel cubic form

In this paper, we study locally strongly convex centroaffine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of centroaffine metric. As the main result, we classify such hypersurfaces being not of flat centroaffine metric. In particular, 2,3-dimensional locally strongly convex centroaffine hypersurfaces with semi-parallel cubic form are completely determined.

Dually flat and projectively flat Minkowskian product Finsler metrics

In this paper, we characterize dually flat (resp. projectively flat) Minkowskian product Finsler metrics, and prove that a Minkowskian product Finsler metric is dually flat and projectively flat if and only if it is a Minkowskian metric.

Exponential arcs in manifolds of quantum states

The manifold under consideration consists of the faithful normal states on a sigma-finite von Neumann algebra in standard form. Tangent planes and approximate tangent planes are discussed. A relative entropy/divergence function is assumed to be given. It is used to generalize the notion of an exponential arc connecting one state to another. The generator of the exponential arc is shown to be unique up to an additive constant. In the case of Araki’s relative entropy, every self-adjoint element of the von Neumann algebra generates an exponential arc. The generators of the composed exponential arcs are shown to add up. The metric derived from Araki’s relative entropy is shown to reproduce the Kubo–Mori metric. The latter is the metric used in linear response theory. The e- and m-connections describe a dual pair of geometries. Any finite number of linearly independent generators determines a submanifold of states connected to a given reference state by an exponential arc. Such a submanifold is a quantum generalization of a dually flat statistical manifold.

Structure of locally conformally flat manifolds satisfying some weakly-Einstein conditions

It is given a complete study of locally conformally flat metrics satisfying some weakly-Einstein conditions. It is shown that they are either a product Mn(c)×Mn(−c) or a warped product R×fRn−1 for some specific warping function. Moreover, some conditions on locally conformally flat fixed points for the RG2 flow are pointed out.

Weak scalar curvature lower bounds along Ricci flow

In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon (2002) that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in the distributional sense provided that the initial metric is W1,p for some n < p ⩽ ∞. As an application, we use this result to study the relation between the Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in the distributional sense, then the manifold is isometric to a Ricci flat manifold.

Rigidity of critical metrics for quadratic curvature functionals

In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals Ft2=∫|Ricg|2dVg+t∫Rg2dVg, t∈R, and S2=∫Rg2dVg. We show that (i) flat surfaces are the only critical points of S2, (ii) flat three-dimensional manifolds are the only critical points of Ft2 for every t>−13, (iii) three-dimensional scalar flat manifolds are the only critical points of S2 with finite energy and (iv) n-dimensional, n>4, scalar flat manifolds are the only critical points of S2 with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M.T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates.

Algebraic-geometry approach to construction of semi-Hamiltonian systems of hydrodynamic type

In this paper, a class of semi-Hamiltonian diagonal systems of hydrodynamic type is constructed using algebraic-geometric methods. For such systems, hydrodynamic integrals and hydrodynamic symmetries are constructed from algebraic-geometric data. Besides, it is described what algebraic-geometric data distinguish in this class Hamiltonian diagonal systems with Hamiltonian structures defined by flat metrics (local Dubrovin-Novikov brackets) and metrics of constant curvature (nonlocal Mokhov-Ferapontov brackets).

On the moduli space of asymptotically flat manifolds with boundary and the constraint equations

On the moduli space of asymptotically flat manifolds with boundary and the constraint equations

ζ-Conformally Flat LP-Kenmotsu Manifolds and Ricci–Yamabe Solitons

In the present paper, we characterize m-dimensional ζ-conformally flat LP-Kenmotsu manifolds (briefly, (LPK)m) equipped with the Ricci–Yamabe solitons (RYS) and gradient Ricci–Yamabe solitons (GRYS). It is proven that the scalar curvature r of an (LPK)m admitting an RYS satisfies the Poisson equation Δr=4(m−1)δ{β(m−1)+ρ}+2(m−3)r−4m(m−1)(m−2), where ρ,δ(≠0)∈R. In this sequel, the condition for which the scalar curvature of an (LPK)m admitting an RYS holds the Laplace equation is established. We also give an affirmative answer for the existence of a GRYS on an (LPK)m. Finally, a non-trivial example of an LP-Kenmotsu manifold (LPK) of dimension four is constructed to verify some of our results.

The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant Y≤0. Previous work by the second and third named authors [10] showed that while the Yamabe flow always converges in a global weighted sense when Y>0, the flow must diverge when Y≤0. We show here in the Y≤0 case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.

On the possibility of a novel (A)dS/CFT relationship emerging in Asymptotic Safety

Quantum Einstein Gravity (QEG), nonperturbatively renormalized by means of a certain asymptotically safe renormalization group (RG) trajectory, is explored by solving its scale dependent effective field equations and embedding the family of emerging 4-dimensional spacetimes into a single 5-dimensional manifold, which thus encodes the complete information about all scales. By construction the latter manifold is furnished with a natural foliation. Heuristically, its leaves are interpreted as physical spacetime observed on different scales of the experimental resolution. Generalizing earlier work on the embedding of d-dimensional Euclidean QEG spacetimes in (d + 1)-dimensional flat or Ricci flat manifolds, we admit Lorentzian signature in this paper and we consider embeddings in arbitrary (d + 1)-dimensional Einstein spaces. Special attention is paid to the sector of maximally symmetric metrics, and the fundamental definition of QEG in d = 4 that employs the cross-over trajectory connecting the non-Gaussian to the Gaussian RG fixed point. Concerning the embedding of the resulting family of 4D de Sitter solutions with a running Hubble parameter, we find that there are only two possible 5D spacetimes, namely the anti-de Sitter manifold AdS5 and the de Sitter manifold dS5. To arrive at this result essential use is made of the monotone scale dependence of the running cosmological constant featured by the gravitational effective average action. We show that if the scale invariance of the QEG fixed points extends to full conformal invariance, the 5D picture of the resulting geometric and field theoretic structure displays a novel kind of “AdS/CFT correspondence”. While strongly reminiscent of the usual string theory-based AdS/CFT correspondence, also clear differences are found.

Energy conditions in extended f(R, G, T) gravity

In this paper, we consider the flat FriedmannLematreRobertson-Walker metric in the presence of perfect fluid models and extended f(R, G, T) gravity (where R is the Ricci scalar, G is the Gauss Bonnet invariant and T stands for trace of energy momentum tensor). In this context, we assume some specific realistic f(R, G, T) models configuration that could be used to explore the finite-time future singularities that arise in late-time cosmic accelerating phases. In this scenario, we choose the most recent estimated values for the Hubble, deceleration, snap and jerk parameters to develop the viability and bounds on the models parameters induced by different energy conditions.

Infinite families of homogeneous Bismut Ricci flat manifolds

Starting from compact symmetric spaces of inner type, we provide infinite families of compact homogeneous spaces carrying invariant non-flat Bismut connections with vanishing Ricci tensor. These examples turn out to be generalized symmetric spaces of order [Formula: see text] and (up to coverings) they can be realized as minimal submanifolds of the Bismut flat model spaces, namely compact Lie groups. This construction generalizes the standard Cartan embedding of symmetric spaces.

Local normal forms of em-wavefronts in affine flat coordinates

In our previous work, we have generalized the notion of dually flat or Hessian manifold to quasi-Hessian manifold; it admits the Hessian metric to be degenerate but possesses a particular symmetric cubic tensor (generalized Amari-Centsov tensor). Indeed, it naturally appears as a singular model in information geometry and related fields. A quasi-Hessian manifold is locally accompanied with a possibly multi-valued potential and its dual, whose graphs are called the $e$-wavefront and the $m$-wavefront respectively, together with coherent tangent bundles endowed with flat connections. In the present paper, using those connections and the metric, we give coordinate-free criteria for detecting local diffeomorphic types of $e/m$-wavefronts, and then derive the local normal forms of those (dual) potential functions for the $e/m$-wavefronts in affine flat coordinates by means of Malgrange's division theorem. This is motivated by an early work of Ekeland on non-convex optimization and Saji-Umehara-Yamada's work on Riemannian geometry of wavefronts. Finally, we reveal a relation of our geometric criteria with information geometric quantities of statistical manifolds.

Birth, transition and maturation of canard cycles in a piecewise linear system with a flat slow manifold

In this work we deal with the canard regime as a part of a canard explosion taking place in a PWL version of the van der Pol equation having a flat critical manifold. The proposed analysis involves the identification of two specific canard cycles, one at the beginning and the other at the end of the canard regime, here called birth and maturation of canards, respectively. Moreover, inside the canard regime, we also analyse the transition from small amplitude canard cycles (canards without head) to large amplitude canard cycles (canards with head) by identifying the maximal canard, transitory canard, and maximum period canard; and then proving that all these cycles are, in fact, different dynamical objects. There have been several works in the classical framework addressing the transitory regime, but from a numerical point of view. Some of these works involve systems exhibiting a flat slow manifold. The flat part of the nullcline implies a different transition from canard cycles without head to those with head than in the classical canard explosion. This is a good choice as a first approximation to the problem because, in particular, the different canard cycles appear further apart from one another. For that reason we have considered a four-zonal PWL system in which the critical manifold in the lateral left linear region is flat.

Subspace foliations and collapse of closed flat manifolds

AbstractWe study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov–Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.

Learning Stable Vector Fields on Lie Groups

Learning robot motions from demonstration requires models able to specify vector fields for the full robot pose when the task is defined in operational space. Recent advances in reactive motion generation have shown that learning adaptive, reactive, smooth, and stable vector fields is possible. However, these approaches define vector fields on a flat Euclidean manifold, while representing vector fields for orientations requires modeling the dynamics in non-Euclidean manifolds, such as Lie Groups. In this paper, we present a novel vector field model that can guarantee most of the properties of previous approaches i.e., stability, smoothness, and reactivity beyond the Euclidean space. In the experimental evaluation, we show the performance of our proposed vector field model to learn stable vector fields for full robot poses as SE(2) and SE(3) in both simulated and real robotics tasks.

$$G_2$$-structures on flat solvmanifolds

In this article we study the relation between flat solvmanifolds and \(G_2\)-geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of \(\mathsf{GL}(n,\mathbb {Z})\) for \(n=5\) and \(n=6\). Then, we look for closed, coclosed and divergence-free \(G_2\)-structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free \(G_2\)-structure whose finite holonomy is cyclic and contained in \(G_2\), and examples of compact flat manifolds admitting a divergence-free \(G_2\)-structure.

Fibering flat manifolds of diagonal type and their fundamental groups

An [Formula: see text]-dimensional closed flat manifold is said to be of diagonal type if the standard representation of its holonomy group [Formula: see text] is diagonal. An [Formula: see text]-dimensional Bieberbach group of diagonal type is the fundamental group of such a manifold. We introduce the diagonal Vasquez invariant of [Formula: see text] as the least integer [Formula: see text] such that every flat manifold of diagonal type with holonomy [Formula: see text] fibers over a flat manifold of dimension at most [Formula: see text] with flat torus fibers. Using a combinatorial description of Bieberbach groups of diagonal type, we give both upper and lower bounds for this invariant. We show that the lower bounds are exact when [Formula: see text] has low rank. We apply this to analyze diffuseness properties of Bieberbach groups of diagonal type. This leads to a complete classification of Bieberbach groups of diagonal type with Klein four-group holonomy and to an application to Kaplansky’s Unit Conjecture.

ON M-PROJECTIVE CURVATURE TENSOR OF PARA-KENMOTSU MANIFOLDS ADMITTING ZAMKOVOY CONNECTION

In this paper, relation between curvature tensors of Levi-Civita connection and Zamkovoy connection on para-Kenmotsu manifolds have been obtained.Quasi M-projectively flat, M-projectively flat and φ-M-projectively flat paraKenmotsu manifolds admitting Zamkovoy connection have been studied. Also, para-Kenmotsu manifolds admitting Zamkovoy connction satisfying M¯ (ξ, U).R¯ = 0 and M¯ (ξ, U).S¯ = 0 have been developed