Civita Connection Research Articles

Cubic Hermite interpolators on the space of probability measures

In this paper, we introduce a novel two‐step modeling method to generalize cubic Hermite interpolators on the space of probability measures . First, we develop new approaches to capture the Riemannian geometric structure of when equipped with Fisher–Rao metric. Furthermore, we develop and detail all numerical tools on , namely, Levi–Civita connection, minimal geodesics, parallel transport, exponential map, and logarithm map. Then, we demonstrate that preliminary analysis results yield significant benefits in constructing an optimal cubic Hermite spline on as a nonlinear Riemannian manifold, precisely where conventional numerical methods fail.

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.

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.

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.

Quantitative evaluation of Frank vector on the basis of continuous field theory of dislocation and differential geometry

This study conducts the modeling and numerical analysis of kink deformation on the basis of continuous field theory of dislocation and differential geometry. In particular, we aim to evaluate the existence and nature of disclination which is expected to be formed at the tip of kink band. Plastic deformation due to dislocation is obtained by numerically solving the Cartan first structure equation. This result yields a Riemann - Cartan manifold which equips Riemannian metric and Levi – Civita connection. By introducing the holonomy, i.e., the integration of curvature on a closed interval, we evaluate the magnitude of Frank vector at the kink tip. The result shows quantitative agreement with those expected from the classical dislocation theory.

(3 + 1)-formulation for gravity with torsion and non-metricity: the stress–energy–momentum equation

We derive the generalized Gauss–Codazzi–Mainardi (GCM) equation for a general affine connection with torsion and non-metricity. Moreover, we show that the metric compatibility and torsionless condition of a connection on a manifold are inherited to the connection of its hypersurface. As a physical application to these results, we derive the (3 + 1)-Einstein field equation (EFE) for a special case of metric-affine -gravity when , the metric-affine general relativity (MAGR). Motivated by the concept of geometrodynamics, we introduce additional variables on the hypersurface as a consequence of non-vanishing torsion and non-metricity. With these additional variables, we show that for MAGR, the energy, momentum, and the stress–energy part of the EFE are dynamical, i.e., all of them contain the derivative of a quantity with respect to the time coordinate. For the Levi–Civita connection, one could recover the Hamiltonian and the momentum (diffeomorphism) constraint, and obtain the standard dynamics of GR.

Einstein Poisson warped product space

In this paper, we provide some results on Poisson manifold (M, Π) with contravariant Levi–Civita connection associated to pair (Π, g). We introduce the notion of Einstein Poisson warped product space (M = B × f F, Π, g f ) (where Π = Π1 + Π2). Moreover, we show that if M is an Einstein Poisson warped product space with nonpositive scalar curvature and compact base B, J 1 is a field endomorphism on T*B satisfies , then M is simply a Riemannian Poisson product. For a contravariant Lorentzian Poisson warped space (M = B × f F, g, Π) (where ) one can determine contravariant Einstein equations and the cosmological constant Λ corresponding to the contravariant Einstein equation G = −Λg. Moreover, it is shown that Einstein equation G = −Λg, induces the contravariant Einstein equation with cosmological constant Λ F on fiber space (F, g F , Π F ).

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

For a compact Lie group G we consider a lattice gauge model given by the G-Hamiltonian system which consists of the cotangent bundle of a power of G with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi–Civita connection of the Killing metric on G. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for $$G = {\mathrm {SU}}(2)$$ the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the—in general singular—symplectically reduced phase space of the corresponding lattice gauge model carries a star product.

On the Non Metrizability of Berwald Finsler Spacetimes

We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi–Civita connection of some Riemannian metric. We show for the first time that this result does not extend to general Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of a metric-affine geometry with vanishing torsion.

Classification of Almost Norden Golden Manifolds

Almost Norden manifolds were classified by means of the Levi–Civita connection by Ganchev and Borisov and by means of the canonical connection by Ganchev and Mihova. This canonical connection was obtained using the potential tensor of the Levi–Civita of the twin metric. We recall that this canonical connection is the well-adapted connection, obtaining its explicit expression for some classes and being able to obtain two equivalent classifications of almost Norden golden manifolds.

Reconstruction and quantization of Riemannian structures

We use algebraic methods to obtain a Cartan-type formula ∇ωη=12(δ(ωη)−(δω)η+ωδη+ω ⊥ dη+dω ⊥ η+d(ω ⊥ η)) for the Levi−Civita connection on a classical Riemannian manifold M in the direction of a 1-form ω (i.e., the usual Levi−Civita connection along the corresponding vector field via the metric). Here, ⊥ denotes a degree −2 bidirectional interior product built from the metric and δ is the divergence or codifferential. We also recover that δ obeys a 7-term relation making the exterior algebra into a Batalin−Vilkovisky algebra. These formulas arise naturally from a novel view of Riemannian structures as cocycles governing the central extension of the classical exterior algebra to a quantum one, motivated by ideas for quantum gravity. The approach also works when the initial exterior algebra is already quantum, allowing us to construct examples of quantum Riemannian structures, including quantum Levi−Civita connections, as cocycle data. Combining with the semidirect product of a differential graded algebra by the quantum differential algebra Ω(t, dt) in one variable, we recover a differential quantization of M×R associated to any conformal Killing vector field on a Riemannian manifold M.

Vortex motion and geometric function theory: the role of connections

We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi–Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections.This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products R × f N 2 and N 1 × f R . Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space.

Flat Connection for Rotating Vacuum Spacetimes in Extended Teleparallel Gravity Theories

Teleparallel geometry utilizes Weitzenböck connection which has nontrivial torsion but no curvature and does not directly follow from the metric like Levi–Civita connection. In extended teleparallel theories, for instance in f ( T ) or scalar-torsion gravity, the connection must obey its antisymmetric field equations. Thus far, only a few analytic solutions were known. In this note, we solve the f ( T , ϕ ) gravity antisymmetric vacuum field equations for a generic rotating tetrad ansatz in Weyl canonical coordinates, and find the corresponding spin connection coefficients. By a coordinate transformation, we present the solution also in Boyer–Lindquist coordinates, often used to study rotating solutions in general relativity. The result hints for the existence of another branch of rotating solutions besides the Kerr family in extended teleparallel gravities.