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On locally symmetric polynomial metrics: Riemannian and Finslerian surfaces

Csaba Vincze, Márk Oláh, Ábris Nagy

Abstract

In the paper we investigate locally symmetric polynomial metrics in special cases of Riemannian and Finslerian surfaces. The Riemannian case will be presented by a collection of basic results (regularity of second root metrics) and formulas up to Gauss curvature. In case of Finslerian surfaces we formulate necessary and sufficient conditions for a locally symmetric fourth root metric in 2D to be positive definite. They are given in terms of the coefficients of the polynomial metric to make checking the positive definiteness as simple and direct as possible. Explicit examples are also presented. The situation is more complicated in case of spaces of dimension more than two. Some necessary conditions and an explicit example are given for a positive definite locally symmetric polynomial metric in 3D. Computations are supported by the MAPLE mathematics software (LinearAlgebra).

On locally symmetric polynomial metrics: Riemannian and Finslerian surfaces

Abstract

In the paper we investigate locally symmetric polynomial metrics in special cases of Riemannian and Finslerian surfaces. The Riemannian case will be presented by a collection of basic results (regularity of second root metrics) and formulas up to Gauss curvature. In case of Finslerian surfaces we formulate necessary and sufficient conditions for a locally symmetric fourth root metric in 2D to be positive definite. They are given in terms of the coefficients of the polynomial metric to make checking the positive definiteness as simple and direct as possible. Explicit examples are also presented. The situation is more complicated in case of spaces of dimension more than two. Some necessary conditions and an explicit example are given for a positive definite locally symmetric polynomial metric in 3D. Computations are supported by the MAPLE mathematics software (LinearAlgebra).
Paper Structure (16 sections, 16 theorems, 137 equations, 1 figure)

This paper contains 16 sections, 16 theorems, 137 equations, 1 figure.

Table of Contents

  1. On the regularity of locally symmetric second root metrics: Riemannian manifolds
  2. Canonical data on Riemannian surfaces
  3. Riemannian surfaces with constant curvature
  4. On the regularity of locally symmetric fourth root metrics: Finsler surfaces
  5. The Hessian
  6. Necessary conditions of positive definiteness
  7. Some examples
  8. The definiteness polynomial
  9. The irreducible case
  10. The reducible case
  11. Equivalent characterization of positive definiteness
  12. Constructing examples
  13. Computations in 3D
  14. Necessary conditions
  15. Computations under a special choice of the coefficients
  16. ...and 1 more sections

Key Result

Corollary 1

OKV A locally symmetric fourth root metric is locally determined by at most five components of its local characteristic polynomial.

Figures (1)

  • Figure 1: Example \ref{['sincosex2d']}: the graph of the coefficient function $n$ (bold curve) with the lower and upper bounds $\frac{3}{2}\sqrt{4l^2+2m^2}-3l$ and $6l$, respectively (dotted curves), and the critical values $\frac{8l^2+m^2}{4l}$ (dashed curve) as functions of $z:=x_1x_2$. The reducibility of the definiteness polynomial changes each time $n$ crosses the dashed curve.

Theorems & Definitions (33)

  • Definition 1
  • Example 1
  • Definition 2
  • Example 2
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Proposition 2
  • Lemma 3
  • ...and 23 more