Finsler geometries on strictly accretive matrices

Billiards in Finsler and Minkowski geometries

We begin the study of billiard dynamics in Finsler geometry. We deduce the Finsler billiard reflection law from the “least action principle”, and extend the basic properties of Riemannian and Euclidean billiards to the Finsler and Minkowski settings, respectively. We prove that the Finsler billiard map is a symplectomorphism, and compute the mean free path of the Finsler billiard ball. For the ...

Geometries on σ-Hermitian matrices

Ring geometry and the geometry of matrices meet naturally at the ring R := Kn×n of n× n matrices with entries in a (not necessarily commutative) field K. Our aim is to strengthen the interaction between these disciplines. Below we sketch some results from either side, even though not in their most general form, but in a way which is tailored for our needs. Let us start with ring geometry, where...

Non-additive Lie centralizer of infinite strictly upper triangular matrices

‎Let $mathcal{F}$ be an field of zero characteristic and $N_{infty‎}(‎mathcal{F})$ be the algebra of infinite strictly upper triangular‎ ‎matrices with entries in $mathcal{F}$‎, ‎and $f:N_{infty}(mathcal{F}‎)rightarrow N_{infty}(mathcal{F})$ be a non-additive Lie centralizer of $‎N_{infty }(mathcal{F})$; that is‎, ‎a map satisfying that $f([X,Y])=[f(X),Y]$‎ ‎for all $X,Yin N_{infty}(mathcal{F})...

Spectral Variation, Normal Matrices, and Finsler Geometry

A Class of Anisotropic (finsler-) Space-time Geometries

A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions (null, spaceor timelike). The metrics are classified according to their group of isometries. These turn out to be isomorphic to subgroups of the Poincaré (Lore...