Deforming convex bodies in Minkowski geometry

We introduce and study certain deformation of Minkowski norms in [Formula: see text] determined by a set of [Formula: see text] linearly independent 1-forms and a smooth positive function of [Formula: see text] variables. In particular, the deformation of a Euclidean norm [Formula: see text] produces a Minkowski norm defined in our recent work; its indicatrix is a rotation hypersurface with a [Formula: see text]-dimensional axis passing through the origin. For [Formula: see text], our deformation generalizes the construction of [Formula: see text]-norms which form a rich class of “computable” Minkowski norms and play an important role in Finsler geometry. We characterize such pairs of a Minkowski norm and its image that Cartan torsions of the two norms either coincide or differ by a [Formula: see text]-reducible term. We conjecture that for [Formula: see text] any Minkowski norm can be approximated by images of a Euclidean norm.

Monte Carlo studies on shape deformation and stability of 3D skyrmions under mechanical stresses

We study the stability/instability of skyrmions under mechanical stresses by Monte Carlo simulations in a 3D disk composed of tetrahedrons. Skyrmions emerge in chiral magnetic materials, such as FeGe and MnSi, under the competition of ferromagnetic interaction (FMI) and Dzyaloshinskii-Moriya interaction (DMI) and are stabilized by the external magnetic field. Recent experimental studies show that skyrmions are also stabilized/destabilized by uniaxial compressive stress perpendicular to or along the magnetic field direction. These phenomena are studied by using a 3D Finsler geometry (FG) model. In this 3D FG model, the DMI coefficient is automatically anisotropic by a geometrically implemented coupling of strains and electronic spins. We find that skyrmions are stabilized (destabilized) by extension (compression) stress along the direction of the applied magnetic field consistent with reported experimental data. This consistency implies that the 3D FG model successfully implements the magnetostrictive or magneto-elastic effect of external mechanical stresses on chiral magnetic orders, including the skyrmion configuration.

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.

On a class of projective Ricci curvature of Finsler metrics

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we investigate the projective Ricci curvature of a class of general [Formula: see text]-metrics satisfying a certain condition, which is invariant under the change of volume form. Moreover, we construct a class of new nontrivial examples on such Finsler metrics.

Weighted projective Ricci curvature in Finsler geometry

In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.