Finsler differential geometry in continuum mechanics: Fundamental concepts, history, and renewed application to ferromagnetic solids

Finsler differential geometry enables enriched mathematical and physical descriptions of the mechanics of materials with microstructure. The first propositions for Finsler geometry in solid mechanics emerged some six decades ago. Ideas set forth in these early works are reviewed, along with subsequent literature culminating in contemporary theories of Finsler-geometric continuum mechanics. Concepts unique to generalized Finsler spaces, in the context of continuum mechanical applications, are highlighted. Capabilities afforded by physical models in generalized Finsler spaces are contrasted with those of standard approaches in affinely connected spaces. Theory and several examples of reduced dimensionality are reported for boundary value problems of fracture and phase transformations, showing how simultaneously novel, physical, and pragmatic model predictions can be obtained from Finsler-type continuum field theory. Lastly, the modern theory is newly applied to describe nonlinear elastic ferromagnetic solids in the magnetically saturated state. A variational approach is used to derive Euler–Lagrange equations for macroscopic and microscopic, i.e., respective electromechanical and electronic continuum, equilibrium states. For a representative generalized Finsler metric depending on material symmetry, augmented conservation laws of macroscopic momentum and electronic spin angular momentum naturally emerge.

Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition

We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.

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 Some Special Finsler Spaces

<p>The present communication has mainly been divided into four sections of which the first section is introductory, the second section deals with R - recurrent of order one. In this section we have derived results telling as to when a - recurrent of order one will be R - recurrent of order one, - recurrent of order one will be a - recurrent of order one. In this section we have also derived the Bianchi’s identity and few more identities which hold in a R - recurrent of order one. The third section deals with R - recurrent of order two. In this section we have observed that the recurrence tensor field is non-symmetric, few more relations and the Bianchi’s identity have been derived in a R - recurrent of order two. In the fourth and the last section we have derived the conditions under which a Landsberg space in a - Finsler space, a - Finsler space is semi - P2- like, a - Finsler space is a - Finsler space, a – Finsler space is P- symmetric, a - Finsler space is P2 like</p>

Homogeneous Finsler spaces with exponential metric

We prove the existence of an invariant vector field on a homogeneous Finsler space with exponential metric, and we derive an explicit formula for the S-curvature of a homogeneous Finsler space with exponential metric. Using this formula, we obtain a formula for the mean Berwald curvature of such a homogeneous Finsler space.

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.

On complete Finsler spaces of constant negative Ricci curvature

Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.