Five-dimensional Manifold Research Articles

Actuation manifold from snapshot data

We propose a data-driven methodology to learn a low-dimensional manifold of controlled flows. The starting point is resolving snapshot flow data for a representative ensemble of actuations. Key enablers for the actuation manifold are isometric mapping as encoder, and a combination of a neural network and a $k$ -nearest-neighbour interpolation as decoder. This methodology is tested for the fluidic pinball, a cluster of three parallel cylinders perpendicular to the oncoming uniform flow. The centres of these cylinders are the vertices of an equilateral triangle pointing upstream. The flow is manipulated by constant rotation of the cylinders, i.e. described by three actuation parameters. The Reynolds number based on a cylinder diameter is chosen to be $30$ . The unforced flow yields statistically symmetric periodic shedding represented by a one-dimensional limit cycle. The proposed methodology yields a five-dimensional manifold describing a wide range of dynamics with small representation error. Interestingly, the manifold coordinates automatically unveil physically meaningful parameters. Two of them describe the downstream periodic vortex shedding. The other three describe the near-field actuation, i.e. the strength of boat-tailing, the Magnus effect and forward stagnation point. The manifold is shown to be a key enabler for control-oriented flow estimation.

Human-Trust Based Feedback Control (HTBFC) Five-Dimensional Manifold Info-Geometric Analysis and how HTBEC Impacts Robotics

This work examines the five-dimensional HTBFC Manifold’s info-geometrics using effective information geometry (IG) techniques. The performance of HTBFC can be analysed from a novel relativistic approach by merging IG framework. Subcases of manifolds in two and three dimensions are studied. Human-machine interactions (HMIs) can now be unified through IG in a novel way because to this study. The presentation includes the essential robotic applications of HTBFC. A few newly identified unsolved issues and suggestions for additional study are included in the conclusions.

Gravitational fields of axially symmetric compact objects in 5D space–time–matter gravity

In the standard Einstein’s theory the exterior gravitational field of any static and axially symmetric stellar object can be described by means of a single function from which we obtain a metric into a four-dimensional space–time. In this work we present a generalization of those so called Weyl solutions to a space–time–matter metric in a five-dimensional manifold within a non-compactified Kaluza–Klein theory of gravity. The arising field equations reduce to those of vacuum Einstein’s gravity when the metric function associated to the fifth dimension is considered to be constant. The calculation of the geodesics allows to identify the existence or not of different behaviours of test particles, in orbits on a constant plane, between the two metrics. In addition, static solutions on the hypersurface orthogonal to the added dimension but with time dependence in the five-dimensional metric are also obtained. The consequences on the variation of the rest mass, if the fifth dimension is identified with it, are studied.

SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a Z3-skew-symmetric covariant SO(3)-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group SO(3)↪SU(5). We find two SO(3)-invariants of Z3-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by K(z,z). We study the invariant properties of K(z,z) and find its stabilizer. Making use of these invariant properties, we define an SO(3)-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an SO(3)-irreducible geometric structure and find its curvature and torsion.

Moving Modulating Pulse and Front Solutions of Permanent Form in a FPU Model with Nearest and Next-to-Nearest Neighbor Interaction

We consider a nonlinear chain of coupled oscillators, which is a direct generalization of the classical Fermi–Pasta–Ulam (FPU) lattice and exhibits, besides the usual nearest neighbor interaction, also next-to-nearest neighbor interaction. For the case of nearest neighbor attraction and next-to-nearest neighbor repulsion we prove that such a lattice admits, in contrast to the classical FPU model, moving modulating front solutions of permanent form, which have small converging tails at infinity and can be approximated by solitary wave solutions of the nonlinear Schrödinger equation. When the associated potentials are even, then the proof yields moving modulating pulse solutions of permanent form, whose profiles are spatially localized. Our analysis employs the spatial dynamics approach as developed by Iooss and Kirchgässner. The relevant solutions are constructed on a five-dimensional center manifold, and their persistence is guaranteed by reversibility arguments.

Some Q-curvature Operators on Five-Dimensional Pseudohermitian Manifolds

We construct Q-curvature operators on d-closed (1, 1)-forms and on $$\overline{\partial }_b$$ -closed (0, 1)-forms on five-dimensional pseudohermitian manifolds. These closely related operators give rise to a new formula for the scalar Q-curvature. As applications, we give a cohomological characterization of CR five-manifolds which admit a Q-flat contact form, and we show that every closed, strictly pseudoconvex CR five-manifold with trivial first real Chern class admits a Q-flat contact form provided the Q-curvature operator on $$\overline{\partial }_b$$ -closed (0, 1)-forms is nonnegative.

Hamiltonian systems on almost cosymplectic manifolds

We determine the Hamiltonian vector field on an odd dimensional manifold endowed with almost cosymplectic structure. This is a generalization of the corresponding Hamiltonian vector field on manifolds with almost transitive contact structures, which extends the contact Hamiltonian systems. Applications are presented to the equations of motion on a particular five-dimensional manifold, the extended Siegel-Jacobi upper-half plane X˜1J. The X˜1J manifold is endowed with a generalized transitive almost cosymplectic structure, an almost cosymplectic structure, more general than transitive almost contact structure and cosymplectic structure. The equations of motion on X˜1J extend the Riccati equations of motion on the four-dimensional Siegel-Jacobi manifold X1J attached to a linear Hamiltonian in the generators of the real Jacobi group G1J(R).

Spinors in Five-Dimensional Contact Geometry

We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely $G_2$ contact geometry and Legendrean contact geometry. The key players in these two geometries are invariantly defined directional derivatives defined only in the contact directions. We explain how to define them and their usage in constructing basic invariants such as the harmonic curvature, the obstruction to being locally flat from the parabolic viewpoint. As an application, we calculate the invariant torsion of the $G_2$ contact structure on the configuration space of a flying saucer (always a five-dimensional contact manifold).

Anatomy of Einstein manifolds

An Einstein manifold in four dimensions has some configuration of $SU(2)_+$ Yang-Mills instantons and $SU(2)_-$ anti-instantons associated with it. This fact is based on the fundamental theorems that the four-dimensional Lorentz group $Spin(4)$ is a direct product of two groups $SU(2)_\pm$ and the vector space of two-forms decomposes into the space of self-dual and anti-self-dual two-forms. It explains why the four-dimensional spacetime is special for the stability of Einstein manifolds. We now consider whether such a stability of four-dimensional Einstein manifolds can be lifted to a five-dimensional Einstein manifold. The higher-dimensional embedding of four-manifolds from the viewpoint of gauge theory is similar to the grand unification of Standard Model since the group $SO(4) \cong Spin(4)/\mathbb{Z}_2 = SU(2)_+ \otimes SU(2)_-/\mathbb{Z}_2$ must be embedded into the simple group $SO(5) = Sp(2)/\mathbb{Z}_2$. Our group-theoretic approach reveals the anatomy of Riemannian manifolds quite similar to the quark model of hadrons in which two independent Yang-Mills instantons represent a substructure of Einstein manifolds.

Extending $$\Lambda (t)-$$CDM to the inflationary epoch using dynamical foliations and a pre-inflationary vacuum energy from 5D geometrical vacuum as a unifying mechanism

In this paper assuming a 5D quantum pre-inflationary vacuum energy, we propose a manner to extend some \(\Lambda (t)\)-CDM models to the inflationary period by using dynamical foliations of the five-dimensional (5D) Ricci-flat space-time manifold, regarding a non-compact extra space-like coordinate. In this formalism, we achieve also a geometrical unification of inflation and the present accelerating epoch. In this approach, inflation is generated by a pre-inflationary quantum vacuum energy that maintains the 5D classical vacuum on cosmological scales. We obtain from geometrical conditions that we can model the presence of the pre-inflationary vacuum energy in 4D as a dynamical cosmological constant. In this model, the 4D inflationary period is governed by a power law expansion and for certain values of some parameters of the model, we obtain an spectral index satisfying \(0.9607\le n_s\le 0.9691\) and a scalar-to-tensor ratio \(r=0.098\), values that fit well according to Planck 2018 results. The 4D inflationary potential is induced for the 5D geometry and the 4D pre-inflationary potential is determined by the model and its contribution is necessary so that \(n_s\) and r can fit the observational data. We also show that in this theoretical framework, the present acceleration in the expansion of the universe can be explained due to a remanent of this pre-inflationary vacuum energy scaled to the present epoch and that its description can be done with the same \(\Lambda (t)\). In this period, we obtain a deceleration parameter in agreement with Planck 2018 data under certain restrictions of the parameters of the model. From the geometrical point of view, \(\Lambda (t)\) is depending on the dynamical foliation of the 5D space-time manifold.

Invariant submanifolds of hyperbolic Sasakian manifolds and η-Ricci-Bourguignon solitons

We set the goal to study the properties of invariant submanifolds of the hyperbolic Sasakian manifolds. It is proven that a three-dimensional submanifold of a hyperbolic Sasakian manifold is totally geodesic if and only if it is invariant. Also, we discuss the properties of ?-Ricci-Bourguignon solitons on invariant submanifolds of the hyperbolic Sasakian manifolds. Finally, we construct a non-trivial example of a three-dimensional invariant submanifold of five-dimensional hyperbolic Sasakian manifold and validate some of our results.

Five-dimensional para-CR manifolds and contact projective geometry in dimension three

We study invariant properties of $5$-dimensional para-CR structures whose Levi form is degenerate in precisely one direction and which are $2$-nondegenerate. We realize that two, out of three, primary (basic) para-CR invariants of such structures are the classical differential invariants known to Monge (1810) and to Wuenschmann (1905) \[ M(G) := 40G_{ppp}^3-45G_{pp}G_{ppp}G_{pppp}+9G_{pp}^2G_{ppppp}, \quad W(H) := 9D^2H_r-27DH_p-18H_rDH_r+18H_pH_r+4H_r^3+54H_z. \] The vanishing $M(G) \equiv 0$ provides a local necessary and sufficient condition for the graph of a function in the $(p,G)$-plane to be contained in a conic, while the vanishing $W(H) \equiv 0$ gives an if-and-only-if condition for a 3rd order ODE to define a natural Lorentzian geometry on the space of its solutions. Mainly, we give a geometric interpretation of the third basic invariant of our class of para-CR structures, the simplest one, of lowest order, and of mixed nature $N(G,H):=2G_{ppp}+G_{pp}H_{rr}$. We establish that the vanishing $N(G,H) \equiv 0$ gives an if-and-only-if condition for the two $3$-dimensional quotients of the para-CR manifold by its two canonical integrable rank-$2$ distributions, to be equipped with contact projective geometries. A curious transformation between the Wuenschmann invariant and the Monge invariant, first noted by us in arxiv.org/abs/2003.08166/, is also discussed, and its mysteries are further revealed.

Holographic inflation in f(T) and DGP braneworld

In this paper, we investigate the holographic inflation in the framework of two gravitational theories like [Formula: see text] gravity, where [Formula: see text] corresponds to the torsion scalar and DGP braneworld model which is based on the idea that our four-dimensional FRW universe located on five-dimensional manifold. The modified infrared cutoff is used to explore all the inflationary parameters as slow-roll parameters [Formula: see text], number of [Formula: see text]-folds [Formula: see text], scalar spectral index [Formula: see text], tensor-to-scalar ratio [Formula: see text] and running of the scalar spectral index [Formula: see text]. Also, we investigate the behavior of inflationary parameters [Formula: see text] and [Formula: see text] through graphical presentation. The behavior of these parameters shows the consistency with Planck data [Formula: see text].

∗-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.

ON f-KENMOTSU MANIFOLDS AND THEIR SUBMANIFOLDS WITH QUARTER SYMMETRIC METRIC CONNECTIONS

The object of the present paper is to study invariant submanifolds of f-Kenmotsu manifolds with respect to quarter symmetric metric connections. Some necessary and sufficient conditions for such submanifolds to be totally geodesic have been deduced. Also we construct an example of a submanifold of a five-dimensional f-Kenmotsu manifold to justify our results.

Decomposable (6, 5)-solutions in 11-dimensional supergravity

This paper presents a series of constructions providing 11-dimensional bosonic supergravity backgrounds. In particular, we treat Lorentzian manifolds given in terms of twisted products of six-dimensional Lorentzian manifolds and five-dimensional Riemannian manifolds. By considering a representative flux 4-form adapted to this setting, we analyse the system of bosonic supergravity equations and describe the corresponding geometric constraints. The new supergravity backgrounds appear for special cases associated to the adapted flux 4-form. For example, we provide a relation of 11-dimensional supergravity with Ricci-isotropic Walker manifolds, and illustrate several results in their terms and in terms of Ricci-flat Riemannian manifolds.

The P'-operator, the Q'-curvature, and the CR tractor calculus

We establish an algorithm which computes formulae for the CR GJMS operators, the P0-operator, and the Q0-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the P0-operator, and shows that the Q0-curvature is constant, with the constant explicitly given in terms of the Webster scalar curvature. We also use our algorithm to derive local formulae for the P0-operator and Q0-curvature of a five-dimensional pseudo-Einstein manifold. Comparison with Marugame’s formulation of the Burns-Epstein invariant as the integral of a pseudohermitian invariant yields new insights into the class of local pseudohermitian invariants for which the total integral is independent of the choice of pseudo-Einstein contact form.

Aerodynamics of Flying Saucers

We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of ‘parabolic geometry’) and we provide examples when this geometry is ‘flat,’ meaning that its symmetries comprise the split form of the exceptional Lie algebra $${\mathfrak {g}}_2$$.

DUALITY OF (2, 3, 5)-DISTRIBUTIONS AND LAGRANGIAN CONE STRUCTURES

As was shown by a part of the authors, for a given $(2,3,5)$-distribution $D$ on a five-dimensional manifold $Y$, there is, locally, a Lagrangian cone structure $C$ on another five-dimensional manifold $X$ which consists of abnormal or singular paths of $(Y,D)$. We give a characterization of the class of Lagrangian cone structures corresponding to $(2,3,5)$-distributions. Thus, we complete the duality between $(2,3,5)$-distributions and Lagrangian cone structures via pseudo-product structures of type $G_{2}$. A local example of nonflat perturbations of the global model of flat Lagrangian cone structure which corresponds to $(2,3,5)$-distributions is given.

Homotopy of gauge groups over non-simply-connected five-dimensional manifolds

Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper, we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity. Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.