[Paper Review] On dually flat general $(\alpha,\beta)$-metrics

Changtao Yu|arXiv (Cornell University)|Dec 31, 2013

Advanced Differential Geometry Research6 references3 citations

TL;DR

This paper establishes a characterization for dually flat general $(\alpha,\beta)$-metrics by deriving a partial differential equation (PDE) that the function $\phi(b^2, s)$ must satisfy, given that the Riemannian metric $\alpha$ is locally dually flat and the 1-form $\beta$ is dually related to $\alpha$. The key contribution is a necessary and sufficient condition for dual flatness in terms of $\phi$, enabling the construction of explicit non-trivial examples via a novel deformation technique.

ABSTRACT

In this work, the dual flatness, which is connected with Statistics and Information geometry, of general $(\\alpha,\\beta)$-metrics (a new class of Finsler metrics) is studied. A nice characterization for such metrics to be dually flat under some suitable conditions is provided and all the solutions are completely determined. By using an original kind of metrical deformations, many non-trivial explicit examples are constructed. Moreover, the relationship of dual flatness and projective flatness of such metrics is shown.

Motivation & Objective

• To characterize dually flat general $(\alpha,\beta)$-metrics under the assumption that $\alpha$ is locally dually flat and $\beta$ is dually related to $\alpha$.
• To provide a necessary and sufficient condition for dual flatness in terms of a PDE involving the function $\phi(b^2, s)$ that defines the metric.
• To construct explicit non-trivial examples of dually flat general $(\alpha,\beta)$-metrics using a new deformation technique.
• To clarify the relationship between dual flatness and projective flatness for general $(\alpha,\beta)$-metrics.

Proposed method

• Derive the PDE condition (1.5): $\phi_2^2 + \phi\phi_{22} + 2s\phi_1\phi_2 + 2s\phi\phi_{12} - 4\phi\phi_1 = 0$ for dual flatness of $F = \alpha\phi(b^2, \beta/\alpha)$, given $\alpha$ and $\beta$ satisfy the duality conditions (1.4).
• Use a deformation technique to construct new examples by solving the PDE for $\phi$ under specific ansatzes for $f(t)$ and $g(t)$, leading to explicit forms of $\phi$.
• Apply the solution structure of the PDE to recover $\phi$ from a related function $\check{\phi}$ satisfying the projective flatness equation (6.1), linking dual flatness and projective flatness.
• Verify that the constructed $\phi$ functions yield valid Finsler metrics by ensuring positivity and smoothness, particularly by requiring $g(b^2) > 0$.
• Use the Hessian structure of $\alpha$ and the dually related condition on $\beta$ to ensure the underlying Riemannian metric and 1-form satisfy the required geometric constraints.
• Establish equivalence between dual flatness and projective flatness by showing that the PDE (1.5) for $\phi$ is equivalent to the projective flatness PDE (6.1) for $\check{\phi} = (\phi^2)_2$.

Experimental results

Research questions

• RQ1 Under what conditions is a general $(\alpha,\beta)$-metric $F = \alpha\phi(b^2, \beta/\alpha)$ dually flat when $\alpha$ is locally dually flat and $\beta$ is dually related to $\alpha$?
• RQ2 What PDE must the function $\phi(b^2, s)$ satisfy for the metric $F$ to be dually flat?
• RQ3 How can one construct explicit non-trivial examples of dually flat general $(\alpha,\beta)$-metrics using a deformation technique?
• RQ4 What is the relationship between dual flatness and projective flatness in the class of general $(\alpha,\beta)$-metrics?
• RQ5 Can the class of dually flat general $(\alpha,\beta)$-metrics be fully characterized via a deformation of dually flat Riemannian metrics and dually related 1-forms?

Key findings

• The general $(\alpha,\beta)$-metric $F = \alpha\phi(b^2, \beta/\alpha)$ is dually flat if and only if the function $\phi$ satisfies the PDE: $\phi_2^2 + \phi\phi_{22} + 2s\phi_1\phi_2 + 2s\phi\phi_{12} - 4\phi\phi_1 = 0$, where $\phi_1$ and $\phi_2$ denote partial derivatives with respect to $b^2$ and $s = \beta/\alpha$.
• Explicit examples of dually flat general $(\alpha,\beta)$-metrics are constructed by choosing specific functions $f(t)$ and $g(t)$, such as $\phi(b^2,s) = \sqrt{g(b^2) + 2g'(b^2)s^2}$, yielding Riemannian metrics when $g$ is positive.
• When $\phi(b^2,s) = \frac{\sqrt{1 - b^2 + s^2} + s}{1 - b^2}$, the metric reduces to a Randers metric, showing that Randers metrics are included in the class of dually flat general $(\alpha,\beta)$-metrics under the given conditions.
• The PDE for dual flatness (1.5) is equivalent to the PDE for projective flatness when $\check{\phi} = (\phi^2)_2$, establishing a direct link between dual flatness and projective flatness in this class.
• The solution structure of the PDE allows for a general parametrization of $\phi$ in terms of $g(b^2)$, $f(t)$, and integrals, enabling systematic construction of new dually flat Finsler metrics.
• For $\phi(b^2,s) = (\sqrt{1+b^2} + s)^{3/2}$, the resulting metric is of Berwald type after scaling, indicating that some dually flat general $(\alpha,\beta)$-metrics are also Berwald metrics.