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Deforming convex bodies in Minkowski geometry
- Publication Year :
- 2019
-
Abstract
- We introduce and study deformation $T_{{\bf b},\phi}$ of Minkowski norms in $\mathbb{R}^n$, determined by a set ${\bf b}=(\beta_1,\ldots,\beta_p)$ of linearly independent 1-forms and a smooth positive function $\phi$ of $p$ variables. In particular, the $T_{{\bf b},\phi}$-image of a Euclidean norm $\alpha$ is a Minkowski norm, whose indicatrix is a rotation hypersurface with a $p$-dimensional axis passing through the origin. For $p=1$, our deformation generalizes construction of $(\alpha,\beta)$-norm; the last ones form a rich class of "computable" Minkowski norms and play an important role in Finsler geometry. We use compositions of $T_{{\bf b},\phi}$-deformations with ${\bf b}$'s of length $p$ to define an equivalence relation $\overset{p}\sim$ on the set of all Minkowski norms in $\mathbb{R}^n$. We apply M. Matsumoto result to characterize the cases when the Cartan torsions of a norm and its $T_{{\bf b},\phi}$-image either coincide or differ by a $C$-reducible term.<br />Comment: 12 pages, 3 figures
- Subjects :
- Mathematics - Differential Geometry
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1910.01854
- Document Type :
- Working Paper