Levi-Civita Research Articles

Parametric approximations of fast close encounters of the planar three-body problem as arcs of a focus-focus dynamics

Abstract A gravitational close encounter of a small body with a planet may produce a substantial change of its orbital parameters which can be studied using the circular restricted three-body problem. In this paper we provide parametric representations of the fast close encounters with the secondary body of the planar CRTBP as arcs of non-linear focus-focus dynamics. The result is the consequence of a remarkable factorisation of the Birkhoff normal forms of the Hamiltonian of the problem represented with the Levi–Civita regularisation. The parameterisations are computed using two different sequences of Birkhoff normalisations of given order N. For each value of N, the Birkhoff normalisations and the parameters of the focus-focus dynamics are represented by polynomials whose coefficients can be computed iteratively with a computer algebra system; no quadratures, such as those needed to compute action-angle variables of resonant normal forms, are needed. We also provide some numerical demonstrations of the method for values of the mass parameter representative of the Sun–Earth and the Sun–Jupiter cases.

Italian and Polish Mathematicians Face Racial Persecution and Emigration: Backgrounds, Individual Fates and Global Aspects

This paper provides a new analytical perspective on the socio-cultural interactions involving members of two main Italian research communities – the School of Mathematical Logic and the Italian School of Algebraic Geometry – and some of their Polish colleagues (Samuel Dickstein, Wiesław Jezierski, Alfred Rosenblatt et al.). I shall argue that the intricate network of relationships between Giuseppe Peano, Corrado Segre and their protégés paved the way for an intensification of scientific exchanges between Italy and Poland in the 1930s. The climax manifested in visits by Tullio Levi-Civita and Mauro Picone to Warsaw and Krakow. Simultaneously, this network fostered instances of remarkable cross-solidarity when Polish and Italian mathematicians faced a period of severe anti-Semitic persecution and migration. An appendix containing unpublished letters documents Rosenblatt’s role in assisting aspiring Italian refugees in securing positions in Peru.

Assuming Photon as Extended Point Particle in the HyperSoft Topological Space and other Hypotheses: Issues and Trend Analysis

Following our preceding article, where we discussed alternative interpretations of the advanced perihelion of Mercury, the present article revisits the 1919 solar eclipse expedition led by Arthur Eddington, which famously provided the first observational confirmation of Einstein's theory of general relativity. We focus on the deflection of starlight data obtained during the eclipse, a cornerstone of this validation. Here, we explore three alternative explanations for the observed light bending that challenge the sole attribution to general relativity. Firstly, the paper begins by arguing based on criticisms raised by Tullio Levi-Civita, a contemporary mathematician, regarding Einstein's use of pseudo-tensors in his calculations. Levi-Civita argued that this approach introduced unnecessary complexity and obscured alternative interpretations of the data. Secondly, we delve into astrophysical phenomena that could mimic the observed light deflection, based on the varying speed of light, by assuming a photon has mass. (cf. ‘t Hooft et al., Light is Heavy). Moreover, in the literature, Molodtsov initiated soft set theory as an extension of fuzzy set theory to deal with uncertainties occurring in the natural and social sciences. It attracted the attention of mathematicians as well as social scientists due to its potential to unify certain mathematical aspects and applications in decision making processes; therefore, we shall discuss a bit how to model the photon as an extended massive particle of light, possibly related to such a soft set point [15, 16]. For further exploration, it is possible to assume the crystalline lattice of subvacuum structure (cf. Gremaud) as part of hypersoft topological spaces [16, 16a]. We strongly believe that the true strength of science lies in its continuous search for evidence and refinement of existing models. Therefore, it can be expected that new data can be helpful to reevaluate these matters, for instance, in the upcoming eclipse in the next month of 2024.

Para-Kähler and pseudo-Kähler structures on Lie–Yamaguti algebras

For a pre-Lie–Yamaguti algebra [Formula: see text], by using its sub-adjacent Lie–Yamaguti algebra [Formula: see text], we are able to construct a semidirect product Lie–Yamaguti algebra via a representation of [Formula: see text]. The investigation of such semidirect Lie–Yamaguti algebras leads us to the notions of para-Kähler structures and pseudo-Kähler structures on Lie–Yamaguti algebras, and also gives the definition of complex product structures on Lie–Yamaguti algebras. Furthermore, a Levi–Civita product with respect to a pseudo-Riemannian Lie–Yamaguti algebra is introduced and we explore its relation with pre-Lie–Yamaguti algebras.

New characterizations of ruled real hypersurfaces in complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections. For any nonnull constant k and any symmetric tensor field of type (1, 1) L on M, we can define two tensor fields of type (1, 2) on M, L_F^{(k)} and L_T^{(k)}, related to both connections. We study the behaviour of the structure operator phi with respect to such tensor fields in the particular case of L=A, the shape operator of M, and obtain some new characterizations of ruled real hypersurfaces in complex projective space.

Pairs of Associated Yamabe Almost Solitons with Vertical Potential on Almost Contact Complex Riemannian Manifolds

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are, in principle, equipped with a pair of mutually associated pseudo-Riemannian metrics. Each of these metrics is specialized as a Yamabe almost soliton with a potential collinear to the Reeb vector field. The resulting manifolds are then investigated in two important cases with geometric significance. The first is when the manifold is of Sasaki-like type, i.e., its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The second case is when the soliton potential is torse-forming, i.e., it satisfies a certain recurrence condition for its covariant derivative with respect to the Levi–Civita connection of the corresponding metric. The studied solitons are characterized. In the three-dimensional case, an explicit example is constructed, and the properties obtained in the theoretical part are confirmed.

Regularized direct method for low–thrust trajectory optimization: Minimum–fuel transfer between cislunar periodic orbits

Low–thrust propulsion systems of high specific impulse are advantageous in the fuel efficiency. The fuel expenditure is further reduced together with optimization techniques though fuel–optimal low–thrust trajectories result in a discontinuous bang–bang control structure that can cause numerical difficulties. The present paper introduces regularized formulations into the direct multiple shooting method of minimizing fuel expenditure for low–thrust spacecraft trajectories. Instantaneous velocity changes approximating low–thrust maneuvers are expressed by regularized variables based on the Levi–Civita or Kustaanheimo–Stiefel transformation. The new formulation removes singularities due to null thrust impulses from derivatives of an objective function and constraints. The method automatically and robustly finds fuel–efficient bang–bang control structures for multi–revolutional transfers between periodic orbits in the Earth–Moon circular restricted three–body problem.

A Dually Flat Embedding of Spacetime

A model of spacetime is presented. It has an extension to five dimensions, and in five dimensions the geometry is the dual of the Euclidean geometry w.r.t. an arbitrary positive-definite metric. Dually flat geometries are well-known in the context of information geometry. The present work explores their role in describing the geometry of spacetime. It is shown that the positive-definite metric with its flat 5-d connection can coexist with a pseudometric for which the connection is that of Levi-Civita. The 4-d geodesics are characterized by five conserved quantities, one of which can be chosen freely and is taken equal to zero in the present work. An explicit expression for the parallel transport operators is obtained. It is used to construct a pseudometric for spacetime by choosing an arbitrary possibly degenerate inner product in the tangent space of a reference point, for instance, that of Minkowski. By parallel transport, one obtains a pseudometric for spacetime, the metric connection of which extends to a 5-d connection with vanishing curvature tensor. The de Sitter space is considered as an example.

English

In this study, we state some basic results about the geometry of the special linear group SL(2,R), seen as a subset of  in terms of the left invariant fields, such as bracketing, Levi Civita connection ∇ and Riemann curvature tensor R, we give some basic theorems for Mannheim partner curves  in the special linear group. We also find the relations between the curvatures and torsions of these associated curves and we give necessary and sufficient conditions for a given curve to be a Mannheim partner curve of another given curve through a relation between its curvature and torsion. Key words: Special linear group, Mannheim curves, Mannheim, partner curves.

First Natural Connection on Riemannian Π-Manifolds

A natural connection with torsion is defined, and it is called the first natural connection on the Riemannian Π-manifold. Relations between the introduced connection and the Levi–Civita connection are obtained. Additionally, relations between their respective curvature tensors, torsion tensors, Ricci tensors, and scalar curvatures in the main classes of a classification of Riemannian Π-manifolds are presented. An explicit example of dimension five is provided.

Bochner–Kodaira Formulas and the Type IIA Flow

A new derivation of the flow of metrics in the Type IIA flow is given. It is better adapted to the formulation of the flow as a variant of a Laplacian flow, and it uses the projected Levi–Civita connection of the metrics themselves instead of their conformal rescalings.

Some pseudo-Kähler Einstein 4-symmetric spaces with a “twin” special almost complex structure

On 4-symmetric symplectic spaces, invariant almost complex structures -up to sign- arise in pairs. We exhibit some 4-symmetric symplectic spaces, with a pair of “natural” compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non-integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non-integrable one is Ricci Hermitian (in the sense that the almost complex structure preserves the Ricci tensor of the associated Levi Civita connection) and special in the sense that the associated Chern Ricci form is proportional to the symplectic form.

On sharp stochastic zeroth-order Hessian estimators over Riemannian manifolds

AbstractWe study Hessian estimators for functions defined over an $n$-dimensional complete analytic Riemannian manifold. We introduce new stochastic zeroth-order Hessian estimators using $O (1)$ function evaluations. We show that, for an analytic real-valued function $f$, our estimator achieves a bias bound of order $ O ( \gamma \delta ^2 ) $, where $ \gamma $ depends on both the Levi–Civita connection and function $f$, and $\delta $ is the finite difference step size. To the best of our knowledge, our results provide the first bias bound for Hessian estimators that explicitly depends on the geometry of the underlying Riemannian manifold. We also study downstream computations based on our Hessian estimators. The supremacy of our method is evidenced by empirical evaluations.

Cartan F(R) Gravity and Equivalent Scalar–Tensor Theory

We investigate the Cartan formalism in F(R) gravity. F(R) gravity has been introduced as a theory to explain cosmologically accelerated expansions by replacing the Ricci scalar R in the Einstein–Hilbert action with a function of R. As is well-known, F(R) gravity is rewritten as a scalar–tensor theory by using the conformal transformation. Cartan F(R) gravity is described based on the Riemann–Cartan geometry formulated by the vierbein-associated local Lorenz symmetry. In the Cartan formalism, the Ricci scalar R is divided into two parts: one derived from the Levi–Civita connection and the other from the torsion. Assuming the spin connection-independent matter action, we have successfully rewritten the action of Cartan F(R) gravity into the Einstein–Hilbert action and a scalar field with canonical kinetic and potential terms without any conformal transformations. red Thus, symmetries in Cartan F(R) gravity are clearly conserved. The resulting scalar–tensor theory is useful in applications of the usual slow-roll scenario. As a simple case, we employ the Starobinsky model and evaluate fluctuations in cosmological microwave background radiation.

Levi–Civita Connections on Quantum Spheres

We introduce q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi–Civita connection for a general class of metrics. We also give metric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.

Note on the Levi-Civita metric

By introducing a new coordinate condition we divide the cylindrical spacetime of Levi-Civita (LC) into a sourceful inner ([Formula: see text]) and vacuum outer ([Formula: see text]) regions. The emergent fluid with zero radial pressure confined to the finely-tuned inner region serves to make the spacetime regular in a natural way. At [Formula: see text] the LC spacetime becomes ‘Cartesian flat’ so that no deficit angle arises.

A Levi–Civita Equation on Monoids, Two Ways

Abstract We consider the Levi–Civita equation f ( x y ) = g 1 ( x ) h 1 ( y ) + g 2 ( x ) h 2 ( y ) f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right) for unknown functions f, g 1, g 2, h 1, h 2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.

Dark energy and accelerating cosmological evolution from osculating Barthel–Kropina geometry

Finsler geometry is an important extension of Riemann geometry, in which each point of the spacetime manifold is associated with an arbitrary internal variable. Two interesting Finsler geometries with many physical applications are the Randers and Kropina type geometries. A subclass of Finsler geometries is represented by the osculating Finsler spaces, in which the internal variable is a function of the base manifold coordinates only. In an osculating Finsler geometry, we introduce the Barthel connection, with the remarkable property that it is the Levi–Civita connection of a Riemannian metric. In the present work we consider the gravitational and cosmological implications of a Barthel–Kropina type geometry. We assume that in this geometry the Ricci type curvatures are related to the matter energy–momentum tensor by the standard Einstein equations. The generalized Friedmann equations in the Barthel–Kropina geometry are obtained by considering that the background Riemannian metric is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equation is also derived. The cosmological properties of the model are investigated in detail, and it is shown that the model admits a de Sitter type solution and that an effective dark energy component can also be generated. Several cosmological solutions are also obtained by numerically integrating the generalized Friedmann equations. A comparison of two specific classes of models with the observational data and with the standard Lambda CDM model is also performed, and it is found that the Barthel–Kropina type models give a satisfactory description of the observations.

Application of regularization maps to quantum mechanical systems in two and three dimensions

In this paper, we generalize the application of the Levi-Civita (L-C) and Kustaanheimo–Stiefel (K-S) regularization methods to quantum mechanical systems in two and three dimensions. Schrödinger equations in two and three dimensions, describing a particle moving under the combined influence of [Formula: see text] and [Formula: see text] potentials are mapped to that of a harmonic oscillator with inverted sextic potential, and interactions, in two and four dimensions, respectively. Using the perturbative solutions of the latter systems, we derive the eigen spectrum of the former systems. Using Bohlin–Sundmann transformation, a mapping between the Schrödinger equations describing shifted harmonic oscillator and H-atom is also derived. Exploiting this equivalence, the solution to the former is obtained from the solution of the latter.

A Levi–Civita functional equation on semigroups

A Levi–Civita functional equation on semigroups