Riemann-Finsler Geometry Research Articles

ON $S$-CURVATURE OF A HOMOGENEOUS FINSLER SPACE WITH RANDERS CHANGED SQUARE METRIC

The study of curvature properties of homogeneous Finsler spaces with $(\alpha, \beta)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on a homogeneous Finsler space with Randers changed square metric has been proved. Further, an explicit formula for $S$-curvature of Randers changed square metric has been established. Finally, using the formula of $S$-curvature, the mean Berwald curvature of afore said $(\alpha, \beta)$-metric has been calculated.

On S-curvature of a homogeneous Finsler space with square metric

The study of curvature properties of homogeneous Finsler spaces with [Formula: see text]-metrics is one of the central problems in Riemann–Finsler geometry. In this paper, the existence of invariant vector fields on a homogeneous Finsler space with square metric is proved. Further, an explicit formula for [Formula: see text]-curvature of a homogeneous Finsler space with square metric is established. Finally, using the formula of [Formula: see text]-curvature, the mean Berwald curvature of aforesaid [Formula: see text]-metric is calculated.

On the Geometry of Some (α, β)-Metrics on the Nilpotent Groups H(p, r)

In this paper we study the Riemann-Finsler geometry of the Lie groups H(p, r) which are a generalization of the Heisenberg Lie groups. For a certain Riemannian metric h·, ·i, the Levi-Civita connection and the sectional curvature are given. We classify all left invariant Randers metrics of Douglas type induced by h·, ·i, compute their flag curvatures and show that all of them are nonBerwaldian.

Almost Analytic Kähler Forms with Respect to a Quadratic Endomorphism with Applications in Riemann-Finsler Geometry

The almost analyticity with respect to a quadratic endomorphism T is introduced in an algebraic setting concerning a commutative and associative algebra \(\mathcal {A}\). Two main properties are proved: the first concerns the simultaneous closedness for an almost analytic 1-form \(\omega \) and \(T\omega \) while the second regards the vanishing of the interior product of such a form with the Nijenhuis tensor of T. Also, we introduce an extension of the Frolicher–Nijenhuis formalism to this framework as well as a hermitian type property. When \(\mathcal {A}\) is the algebra of smooth functions on a given (even dimensional) manifold we recover the classical notion of almost analytic 1-form. We study this analyticity and the hermitian type property for the Cartan 1-form of a Riemann-Finsler geometry. Also, we study the almost analytic functions on the tangent bundle of a Riemann-Finsler geometry with respect to the associated almost para-complex and almost complex structure of this geometry. We introduce two new types of Hessian and respectively Laplacian corresponding to these structures. Two types of gradient Ricci solitons are introduced in the tangent bundle.

CR-structures of codimension 2 on tangent bundles in Riemann–Finsler geometry

We determine a 2-codimensional CR-structure on the slit tangent bundle $$T_0M$$ of a Finsler manifold (M, F) by imposing a condition on the almost complex structure $$\Psi $$ associated to F when restricted to the structural distribution of a framed f-structure. This condition is satisfied when (M, F) is of scalar flag curvature (particularly flat). In the Riemannian case (M, g) this last condition means that g is of constant curvature. This CR-structure is finally generalized by using one positive parameter but under more difficult conditions.

Chern forms of holomorphic Finsler vector bundles and some applications

In this paper, we present two kinds of total Chern forms [Formula: see text] and [Formula: see text] as well as a total Segre form [Formula: see text] of a holomorphic Finsler vector bundle [Formula: see text] expressed by the Finsler metric [Formula: see text], which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133–144] to some extent. As some applications, we show that the signed Segre forms [Formula: see text] are positive [Formula: see text]-forms on [Formula: see text] when [Formula: see text] is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler–Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou’s one [Finsler geometry on complex vector bundles, in A Sampler of Riemann–Finsler Geometry, MSRI Publications, Vol. 50 (Cambridge University Press, 2004), pp. 83–105] and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.

Several problems in Riemann-Finsler geometry

In differential geometry, Finsler geometry is more general than Riemannian geometry, which has been developed essentially in recent decades. In this paper, some open problems in global Finsler geometry are summarized briefly. Such problems deal with Finsler submanifolds, the curvature and topology of Finsler manifolds, and Finsler-Einstein metrics.

A Simplified Algorithm for Inverting Higher Order Diffusion Tensors

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann–Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices.

ON THE RIEMANN CURVATURE OPERATORS IN RANDERS SPACES

The Riemann curvature in Riemann–Finsler geometry can be regarded as a collection of linear operators on the tangent spaces. The algebraic properties of these operators may be linked to the geometry and the topology of the underlying space. The principal curvatures of a Finsler space (M, F) at a point x are the eigenvalues of the Riemann curvature operator at x. They are real functions κ on the slit tangent manifold TM0. A principal curvature κ(x, y) is said to be isotropic (respectively, quadratic) if κ(x, y)/F(x, y) is a function of x only (respectively, κ(x, y) is quadratic with respect to y). On the other hand, the Randers metrics are the most popular and prominent metrics in pure and applied disciplines. Here, it is proved that if a Randers metric admits an isotropic principal curvature, then F is of isotropic S-curvature. The same result is also established for F to admit a quadratic principal curvature. These results extend Shen's verbal results about Randers metrics of scalar flag curvature K = K(x) as well as those Randers metrics with quadratic Riemann curvature operator. The Riemann curvature [Formula: see text] may be broken into two operators [Formula: see text] and [Formula: see text]. The isotropic and quadratic principal curvature are characterized in terms of the eigenvalues of [Formula: see text] and [Formula: see text].

Bipartite Riemann–Finsler geometry and Lorentz violation

Bipartite Riemann–Finsler geometries with complementary Finsler structures are constructed. Calculable examples are presented based on a bilinear-form coefficient for explicit Lorentz violation.

ON A CLASS OF PROJECTIVELY FLAT FINSLER METRICS OF NEGATIVE CONSTANT FLAG CURVATURE

In this paper, we prove a structure theorem for projectively flat Finsler metrics of negative constant flag curvature. We show that for such a Finsler metric if the orthogonal group acts as isometries, then the Finsler metric is a slight generalization of Chern–Shen's construction Riemann–Finsler geometry, Nankai Tracts in Mathematics, Vol. 6 (World Scientific Publishing, Hackensack, NJ, 2005), x+192 pp.

Riemann–Finsler geometry and Lorentz-violating kinematics

Effective field theories with explicit Lorentz violation are intimately linked to Riemann–Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann–Finsler spacetimes and Riemann–Finsler spaces. An example is presented that is constructed from a 1-form coefficient and has Finsler structure complementary to the Randers structure.

Finsler Streamline Tracking with Single Tensor Orientation Distribution Function for High Angular Resolution Diffusion Imaging

We introduce a new framework based on Riemann-Finsler geometry for the analysis of 3D images with spherical codomain, more precisely, for which each voxel contains a set of directional measurements represented as samples on the unit sphere (antipodal points identified). The application we consider here is in medical imaging, notably in High Angular Resolution Diffusion Imaging (HARDI), but the methods are general and can be applied also in other contexts, such as material science, et cetera, whenever direction dependent quantities are relevant. Finding neural axons in human brain white matter is of significant importance in understanding human neurophysiology, and the possibility to extract them from a HARDI image has a potentially major impact on clinical practice, such as in neuronavigation, deep brain stimulation, et cetera. In this paper we introduce a novel fiber tracking method which is a generalization of the streamline tracking used extensively in Diffusion Tensor Imaging (DTI). This method is capable of finding intersecting fibers in voxels with complex diffusion profiles, and does not involve solving extrema of these profiles. We also introduce a single tensor representation for the orientation distribution function (ODF) to model the probability that a vector corresponds to a tangent of a fiber. The single tensor representation is chosen because it allows a natural choice of Finsler norm as well as regularization via the Laplace-Beltrami operator. In addition we define a new connectivity measure for HARDI-curves to filter the most prominent fiber candidates. We show some very promising results on both synthetic and real data.

Naturally reductive homogeneous Finsler spaces

In this paper, we study naturally reductive Finsler metrics. We first give a sufficient and necessary condition for a Finsler metric to be naturally reductive with respect to certain transitive group of isometries. Then we study in detail the left invariant naturally reductive metrics on compact Lie groups and give a method to construct the non-Riemannian ones. Further, we give a classification of left invariant naturally reductive metrics on nilpotent Lie groups. Finally, we give a classification of all the naturally reductive Finsler spaces of dimension less or qual to 4. As applications, we obtain some rigidity theorems about naturally reductive Finsler metrics. Namely, any left invariant non-symmetric naturally reductive Finsler metric on a compact simple Lie group or an indecomposable nilpotent Lie group must be Riemannian. On the other hand, we provide a very convenient method to construct non-symmetric Berwald spaces which are neither Riemannian nor locally Minkowskian, a kind of spaces which are sought after in the book by Bao et al. (An introduction to Riemann–Finsler geometry, GTM 200, 2000).

Curve flows in Lagrange–Finsler geometry, bi-Hamiltonian structures and solitons

Methods in Riemann–Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections ( N -connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces G / S O ( n ) provided with an N -connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrödinger maps on a tangent bundle.

Optimal Placement of a Deposit between Markets: Riemann-Finsler Geometrical Approach

By using the Riemann-Finsler geometry, we study the existence and location of the optimal points for a general cost function involving Finsler distances. Our minimization problem provides a model for the placement of a deposit within a domain with several markets such that the total transportation cost is minimal. Several concrete examples are studied either by precise mathematical tools or by evolutionary (computer assisted) techniques.

Riemann-Finsler geometry

In this paper, a survey on Riemann-Finsler geometry is given. Non-trivial examples of Finsler metrics satisfying different curvature conditions are presented. Local and global results in Finsler geometry are analyzed.

Riemann-Finsler Geometry with Applications to Information Geometry

Information geometry is a new branch in mathematics, originated from the applications of differential geometry to statistics. In this paper we briefly introduce Riemann-Finsler geometry, by which we establish Information Geometry on a much broader base, so that the potential applications of Information Geometry will be beyond statistics.

ELECTROMAGNETIC GAUSSIAN BEAMS AND RIEMANNIAN GEOMETRY

A Gaussian beam is an asymptotic solution to Maxwell’s equations that propagate along a curve; at each time instant its energy is concentrated around one point on the curve. Such a solution is of the form E = Re{eE0(x, t)}, where E0 is a complex vector field, P > 0 is a big constant, and θ is a complex second order polynomial in coordinates adapted to the curve. In recent work by A. P. Kachalov, electromagnetic Gaussian beams have been studied in a geometric setting. Under suitable conditions on the media, a Gaussian beam is determined by Riemann-Finsler geometry depending only on the media. For example, geodesics are admissible curves for Gaussian beams and a curvature equation determines the second order terms in θ. This work begins with a derivation of the geometric equations for Gaussian beams following the work of A. P. Kachalov. The novel feature of this work is that we characterize a class of inhomogeneous anisotropic media where the induced geometry is Riemannian. Namely, if e, μ are simultaneously diagonalizable with eigenvalues ei, μj , the induced geometry is Riemannian if and only if eiμj = ejμi for some i = j. What is more, if the latter condition is not met, the geometry is ill-behaved. It is neither smooth nor convex. We also calculate Riemannian metrics for different media. In isotropic media, gij = eμ δij and in more complicated media there are two Riemannian metrics due to different polarizations.

Volume Comparison and Its Applications in Riemann–Finsler Geometry

We introduce a new geometric quantity,the mean covariationfor Finsler metrics, and establish a volume comparison theorem. As an application, we obtain some precompactness and finiteness theorems for Finsler manifolds.