[Paper Review] On Generalized m-th Root Finsler Metrics

Akbar Tayebi, E. Peyghan+1 more|arXiv (Cornell University)|Feb 11, 2013

Advanced Differential Geometry Research6 references3 citations

TL;DR

This paper investigates generalized $m$-th root Finsler metrics, characterizing locally dually flat structures via a system of PDEs involving a 1-form $\theta$. It proves that if such a metric is projectively related to a standard $m$-th root metric, then either $B=0$ or the metrics reduce to Riemannian ones. The key result is that conformal invariance between a generalized $m$-th root metric and a standard $m$-th root metric forces both to be Riemannian, generalizing known results on Douglas metrics.

ABSTRACT

In this paper, we characterize locally dually flat generalized m-th root Finsler metrics. Then we find a condition under which a generalized m-th root metric is projectively related to a m-th root metric. Finally, we prove that if a generalized m-th root metric is conformal to a m-th root metric, then both of them reduce to Riemannian metrics.

Motivation & Objective

• To characterize locally dually flat generalized $m$-th root Finsler metrics using a system of partial differential equations.
• To determine conditions under which a generalized $m$-th root metric is projectively related to a standard $m$-th root metric.
• To investigate the consequences of conformal equivalence between generalized $m$-th root metrics and standard $m$-th root metrics.
• To generalize known results on Douglas metrics and their reduction to Berwald or Riemannian structures.

Proposed method

• Derive the fundamental metric tensor $g_{ij}$ for the generalized $m$-th root Finsler metric $F = \sqrt{A^{2/m} + B}$ using second-order derivatives of $A$ and $B$.
• Use the condition for local duality flatness: $(F^2)_{x^k y^l} y^k = 2(F^2)_{x^l}$, and express it in terms of $A_{x^l}$, $B_{0l}$, and a 1-form $\theta$.
• Establish projective equivalence between $\bar{F} = \sqrt{A^{2/m} + B}$ and $F = A^{1/m}$ by analyzing the spray coefficients $G^i$ and $\bar{G}^i$ under a scalar function $P(x,y)$.
• Apply conformal change $\bar{F} = e^\alpha \tilde{F}$ and derive the relation $g_{ij} = \frac{1}{1 - e^{2\alpha}}(e^{2\alpha} \tilde{b}_{ij} - \bar{b}_{ij})$ to analyze the metric's Riemannian nature.
• Use tensor identities involving $A_{ij}$, $A^{ij}$, and $y^i$ to simplify curvature and spray coefficient expressions.
• Employ contradiction arguments by assuming non-Riemannian structure and deriving $c_s = 0$ from non-degeneracy conditions, leading to $B = 0$.

Experimental results

Research questions

• RQ1 Under what conditions is a generalized $m$-th root Finsler metric locally dually flat?
• RQ2 When is a generalized $m$-th root metric projectively related to a standard $m$-th root metric?
• RQ3 What happens to the geometry of a generalized $m$-th root metric under conformal equivalence to a standard $m$-th root metric?
• RQ4 Can a non-Riemannian generalized $m$-th root metric be conformally equivalent to a standard $m$-th root metric?

Key findings

• A generalized $m$-th root Finsler metric $F = \sqrt{A^{2/m} + B}$ is locally dually flat if and only if there exists a 1-form $\theta$ such that $B_{0l} = 2B_{x^l}$ and $A_{x^l} = \frac{1}{3m}[mA\theta_l + 2\theta A_l]$.
• If a generalized $m$-th root metric $\bar{F} = \sqrt{A^{2/m} + B}$ is projectively related to $F = A^{1/m}$, then under condition (7), $\bar{F}$ is projectively related to $F$; if condition (8) holds, then $B = 0$, so $\bar{F} = F$.
• If a generalized $m$-th root metric $\bar{F}$ is conformally equivalent to a standard $m$-th root metric $F = A^{1/m}$, then $F$ must be Riemannian, i.e., $C_{ijk} = 0$.
• Conformal equivalence between two generalized $m$-th root metrics $\bar{F} = \sqrt{A^{2/m} + \bar{B}}$ and $\tilde{F} = \sqrt{A^{2/m} + \tilde{B}}$ implies $\bar{B} = \tilde{B}$ if $F = A^{1/m}$ is not Riemannian.
• If $\bar{F}$ is conformal to $F = A^{1/m}$ and $F$ is not Riemannian, then $\bar{F} = F$, so no nontrivial conformal deformation exists.
• The proof shows that non-vanishing $d^j c_j$ and non-degeneracy of curvature terms force $c_s = 0$, leading to $B = 0$, thus collapsing the metric to the standard $m$-th root case.