Finsler Structure Research Articles

Complexity for quantum field theory states and applications to thermofield double states

This paper defines a complexity between states in quantum field theory by introducing a Finsler structure based on ladder operators (the generalization of creation and annihilation operators). Two simple models are shown as examples to clarify the differences between complexity and other conceptions such as complexity of formation and entanglement entropy. When it is applied into thermofield double (TFD) states in $d$-dimensional conformal field theory, results show that the complexity density between them and corresponding vacuum states are finite and proportional to $T^{d-1}$, where $T$ is the temperature of TFD state. Especially, a proof is given to show that fidelity susceptibility of a TFD state is equivalent to the complexity between it and corresponding vacuum state, which gives an explanation why they may share the same object in holographic duality. Some enlightenments to holographic conjectures of complexity are also discussed.

Wind Riemannian spaceforms and Randers–Kropina metrics of constant flag curvature

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to obtain the local classification. For the global one, a suitable result on completeness for WRS yields the complete simply connected models. In particular, any of the local models in the Randers classification does admit an extension to a unique model of wind Riemannian structure, even if it cannot be extended as a complete Finslerian manifold. Thus, WRS's emerge as the natural framework for the analysis of Randers spaceforms and, prospectively, wind Finslerian structures would become important for other global problems too. For the sake of completeness, a brief overview about WRS (including a useful link with the conformal geometry of a class of relativistic spacetimes) is also provided.

Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle $$T_{\mathbb {C}}E=T'E\oplus T''E$$ and its link, via the tangent anchor map, with the complexified tangent bundle $$T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)$$ . A holomorphic Lie algebroid structure is emphasized on $$T'E$$ . A special study is made for integral curves of a spray on $$T'E$$ . Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on $$T'E$$ . In the second part of section two, we study the holomorphic prolongation $$\mathcal {T}'E$$ of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on $$T'M$$ induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from $$T'M$$ .

Intrinsic geometry and analysis of Finsler structures

In this short note, we prove that if F is a weak upper semicontinuous admissible Finsler structure on a domain in $$\mathbb {R}^n, n\ge 2$$ , then the intrinsic distance and differential structures coincide.

Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

Suppose (X,ω) is a compact Kähler manifold. We introduce and explore the metric geometry of the Lp,q-Calabi Finsler structure on the space of Kähler metrics H. After noticing that the Lp,q-Calabi and Lp′-Mabuchi path length topologies on H do not typically dominate each other, we focus on the finite entropy space EEnt, contained in the intersection of the Lp-Calabi and L1-Mabuchi completions of H and find that after a natural strengthening, the Lp-Calabi and L1-Mabuchi topologies coincide on EEnt. As applications to our results, we give new convergence results for the Kähler–Ricci flow and the weak Calabi flow.

The Finslerian wormhole models

We present models of wormhole under the Finslerian structure of spacetime. This is a sequel of our previous work (Eur Phys J 75:564, 2015) where we constructed a toy model for compact stars based on the Finslerian spacetime geometry. In the present investigation, a wide variety of solutions are obtained that explore wormhole geometry by considering different choices for the form function and energy density. The solutions, like the previous work, are revealed to be physically interesting and viable models for the explanation of wormholes as far as the background theory and literature are concerned.

Adapted metrics and Webster curvature in Finslerian 2-dimensional geometry

The Webster scalar curvature is computed for the sphere bundle T1S of a Finsler surface (S, F) subject to the Chern-Hamilton notion of adapted metrics. As an application, it is derived that in this setting (T1S, gSasaki) is a Sasakian manifold homothetic with a generalized Berger sphere, and that a natural Cartan structure is arising from the horizontal 1-forms and the author associates a non-Einstein pseudo-Hermitian structure. Also, one studies when the Sasaki type metric of T1S is generally adapted to the natural co-frame provided by the Finsler structure.

Entropy of foliations with leafwise Finsler structure

We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of a foliated manifold with leafwise Randers structure is considered. In this case the estimates for one dimensional foliation defined by a vector field in terms of the topological entropy of a flow are presented.

The Finslerian compact star model

We construct a toy model for compact stars based on the Finslerian structure of spacetime. By assuming a particular mass function, we find an exact solution of the Finsler-Einstein field equations with an anisotropic matter distribution. The solutions are revealed to be physically interesting and pertinent for the explanation of compact stars.

A Differential Lyapunov Framework for Contraction Analysis

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves.

Paracontact Finsler Structures on Vector Bundles

In this paper, we study the almost paracontact, almost paracontact metric, paracontact metric, K-paracontact and para-Sasakian Finsler structures on vector bundles and give some characterizations for these geometric structures. Also, the curvature of a paracontact Finsler manifold is given and some results for Ricci semi-symmetric para-Sasakian Finsler manifolds and para-Sasakian Finsler manifolds with η-parallel Ricci tensor are obtained with the aid of Ricci tensor and scalar curvature of Finsler structure.

Generalized Finsler structures on closed 3-manifolds

An $(I,J,K)$-generalized Finsler structure on a 3-manifold is a generalization of a Finslerian structure, introduced by R. Bryant in order to separate and clarify the local and global aspects in Finsler geometry making use of Cartan's method of exterior differential systems. In this paper, we show that there is a close relation between $(I,J,1)$-generalized Finsler structures and a class of contact circles, namely the so-called Cartan structures.This correspondence allows us to determine the topology of 3-manifolds that admit $(I,J,1)$-generalized Finsler structures and to single out classes of $(I,J,1)$-generalized Finsler structures induced by standard Cartan structures.

On the Finsler structure of Teichmüller’s metric and Thurston’s metric

There are several important metrics on Teichmüller space. One of them is the classical Teichmüller metric, introduced by Teichmüller in 1939, and another one is Thurston’s asymmetric metric, introduced by Thurston in 1986. Thurston defined his asymmetric metric in analogy with Teichmüller’s metric, as a solution to an extremal problem. Whereas the Teichmüller distance between two points is defined as the result of searching for the best quasiconformal constant for a map between two conformal structures in a given homotopy class, the Thurston distance is defined as the result of searching for the best Lipschitz constant of a homeomorphism between two hyperbolic structures in a given homotopy class. It turns out that the ideas underlying some known properties of Thurston’s asymmetric metric can be used to get new insight into Teichmüller’s metric, and vice versa. The aim of this paper is to highlight several analogies between the Finsler (infinitesimal) properties of these two metrics. In this direction, in analogy with Thurston’s formula for the Finsler norm of a vector with respect to the asymmetric metric, we give a new formula for the Finsler norm of a vector for Teichmüller’s metric. Whereas the former uses the hyperbolic length function, the latter uses the extremal length function. We also describe an embedding of projective measured foliation space in the cotangent space to Teichmüller space. The image of this embedding is the boundary of the dual of the unit ball for the Finsler structure associated to Teichmüller’s metric.

Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions

In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete C k C^{k} Finsler manifold M M is determined by the normed algebra C b k ( M ) C_b^k(M) of all real-valued, bounded and C k C^k smooth functions with bounded derivative defined on M M . As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete C k C^{k} Finsler manifold M M is determined by the algebra C b k ( M ) C_b^k(M) ; (ii) the weak Finsler structure of a separable and complete C k C^{k} Finsler manifold M M modeled on a Banach space with a Lipschitz and C k C^k smooth bump function is determined by the algebra C b k ( M ) C^k_b(M) ; (iii) the weak Finsler structure of a C 1 C^1 uniformly bumpable and complete C 1 C^{1} Finsler manifold M M modeled on a Weakly Compactly Generated (WCG) Banach space is determined by the algebra C b 1 ( M ) C^1_b(M) ; and (iv) the isometric structure of a WCG Banach space X X with a C 1 C^1 smooth bump function is determined by the algebra C b 1 ( X ) C_b^1(X) .

Finslerian extensions of geodesic active fields for digital image registration

AbstractThe present work extends the already established prolific geodesic active field methodology of Riemannian type, which generalizes the Casseles‐Kimmel‐Sapiro weighted length problem, and a direction dependent Finslerian structure is proposed as alternative for the GAF image registration process. The proposed extension relies on the direction‐dependent 0‐homogeneous associated weight metric tensor and aims to achieve the minimization of the absolute error in the case of motion detection. (© 2013 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Convex bodies in Euclidean and Weil-Petersson geometries

On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting metric structure shares with Thurston's asymmetric metric is noted.

Relativistic Finsler geometry

We briefly review some basic concepts of parallel displacement in Finsler geometry. In general relativity, the parallel translation of objects along the congruence of the fundamental observer corresponds to the evolution in time. By dropping the quadratic restriction on the measurement of an infinitesimal distance, the geometry is generalized to a Finsler structure. Apart from curvature, a new property of the manifold complicates the geometrodynamics, the color. The color brings forth an intrinsic local anisotropy, and many quantities depend on position and to a ‘supporting’ direction. We discuss this direction dependence and some physical interpretations. Also, we highlight that in Finsler geometry the parallel displacement is not necessarily always along the ‘supporting’ direction. The latter is a fundamental congruence of the manifold and induces a natural 1 + 3 decomposition. Its internal deformation is given through the evolution of the irreducible components of vorticity, shear, and expansion. Copyright © 2013 John Wiley &amp; Sons, Ltd.

ON FINSLERIZED ABSOLUTE PARALLELISM SPACES

The aim of this paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism (GAP)-space. The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized absolute parallelism (parallelizable) space, combines within its geometric structure the simplicity of GAP-geometry and the richness of Finsler geometry, hence is potentially more suitable for applications and especially for describing physical phenomena. A study of the geometry of the two structures and their interrelation is carried out. Five connections are introduced and their torsion and curvature tensors derived. Some special Finslerized parallelizable spaces are singled out. One of the main reasons to introduce this new space is that both absolute parallelism and Finsler geometries have proved effective in the formulation of physical theories, so it is worthy to try to build a more general geometric structure that would share the benefits of both geometries.

Finsler geometric perspective on the bulk flow in the universe

Astronomical observations showed that there may exist a bulk flow with peculiar velocities in the universe, which contradicts with the ΛCDM model. The bulk flow reveals that the observational universe is anisotropic at large scales. In fact, a more reliable observation on the anisotropy of spacetime comes from the CMB power spectra. The WMAP and Planck satellites both show that there is a hemispherical power asymmetry at large-angular scales. In this Letter, we propose a “wind” scenario to the bulk flow (or the anisotropy of spacetime). Under the influence of the “wind”, the spacetime metric could become a Finsler structure. By resolving the null geodesic equation, we obtain the modified luminosity distance, which has a dipolar form at the leading order. Thus, the “wind” describes well the bulk flow. In addition, we perform a least-χ2 fit to the data of type Ia supernovae (SNe Ia) in the Union2.1 compilation. The peculiar velocity of the bulk flow has an upper limit vbulk≲4000 km/s, which is compatible with all the existing observational values.

Sasakian Finsler manifolds

In this study, almost contact Finsler structures on vector bundle are defined and the condition of normality in terms of the Nijenhuis torsion N_{\phi} of almost contact Finsler structure is obtained. It is shown that for a K-contact structure on Finsler manifold \nabla_X \xi =-\frac{1}{2} \phi X and the flag curvature for plane sections containing \xi are equal to \frac{1}{4}. By using the Sasakian Finsler structure, the curvatures of a Finsler connection \nabla on V are obtained. We prove that a locally symmetric Finsler manifold with K-contact Finsler structure has a constant curvature \frac{1}{4}. Also, the Ricci curvature on Finsler manifold with K-contact Finsler structure is given. As a result, Sasakian structures in Riemann geometry and Finsler condition are generalized. As a conclusion we can state that Riemannian Sasakian structures are compared to Sasakian Finsler structures and it is proven that they are adaptable.