[Paper Review] On Douglas general $(α,β)$-metrics

Xiaoming Wang, Benling Li|arXiv (Cornell University)|Jun 26, 2016

Advanced Differential Geometry Research7 references3 citations

TL;DR

This paper characterizes Douglas general $(\alpha,\beta)$-metrics by deriving a system of PDEs that determine when such Finsler metrics have vanishing Douglas curvature. It provides a complete solution by solving these PDEs, leading to the construction of new non-Riemannian Douglas metrics through explicit functions $\phi(b^2,s)$ and conditions on the covariant derivative of $\beta$. The key contribution is a general classification of Douglas metrics in the broader class of general $(\alpha,\beta)$-metrics without restrictive assumptions like $\beta$ being closed or conformal.

ABSTRACT

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α=\sqrt{a_{ij}(x)y^iy^j}$ and a $1$-form $β=b_i(x)y^i$. We obtain the differential equations that characterizes these metrics with vanishing Douglas curvature. By solving the equivalent PDEs, the metrics in this class are totally determined. Then many new Douglas metrics are constructed.

Motivation & Objective

• To extend the classification of Douglas metrics beyond the restricted cases where $\beta$ is closed or conformal with respect to $\alpha$.
• To characterize all non-Riemannian general $(\alpha,\beta)$-metrics with vanishing Douglas curvature in dimensions $n \geq 3$.
• To derive a complete system of differential equations that fully determine such Douglas metrics.
• To construct explicit new examples of Douglas metrics by solving the derived PDEs and specifying functions $\phi(b^2,s)$, $k(x)$, $c(b^2)$, $\mu(b^2)$, and $\nu(b^2)$.

Proposed method

• Derive the necessary and sufficient conditions for a general $(\alpha,\beta)$-metric $F = \alpha \phi(b^2, s)$ with $s = \beta/\alpha$ to be a Douglas metric by analyzing its Douglas curvature.
• Establish a system of two equations: a second-order PDE in $\phi$ (Equation 1.3) and a condition on the covariant derivative $b_{i|j}$ (Equation 1.4) involving functions $k(x)$, $c(b^2)$, $\mu(b^2)$, and $\nu(b^2)$.
• Solve the PDE (1.3) using a change of variables $\zeta(b^2, s)$ and express $\phi$ in terms of an arbitrary smooth function $h(b^2)$ and a specific rational-trigonometric-exponential expression.
• Use the geodesic coefficient formula $G^i = \hat{G}^i + k\alpha \{[(1-c)s^2 + cb^2]\Theta + b^2\Xi\}y^i$ to relate the spray coefficients of $F$ to those of $\alpha$, with $\hat{G}^i$ and $\Theta, \Xi$ defined explicitly.
• Construct explicit examples by choosing specific forms of $c(b^2)$, $\mu(b^2)$, $\nu(b^2)$, and $\Phi(\zeta)$, leading to new Douglas metrics such as $\phi(b^2,s) = h(b^2)s + \text{complex rational-trigonometric expression}$.
• Provide special solutions for $b_{i|j}$ when $\alpha$ is projectively flat (constant curvature) or conformal to the Euclidean metric $|y|$, yielding concrete examples of $\beta$ and $k(x)$ satisfying (1.4).

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a general $(\alpha,\beta)$-metric $F = \alpha \phi(b^2, \beta/\alpha)$ to be a Douglas metric when $\beta$ is not parallel to $\alpha$?
• RQ2 How can the Douglas curvature condition be reduced to a system of PDEs involving $\phi$, $b^2$, $s$, and the covariant derivative $b_{i|j}$?
• RQ3 Can the PDE in $\phi$ (Equation 1.3) be solved explicitly to generate new non-Riemannian Douglas metrics?
• RQ4 What are the structural forms of $\beta$ and $k(x)$ that satisfy the covariant derivative condition (1.4) in the absence of the conformal or closedness assumptions?
• RQ5 How can explicit examples of Douglas metrics be constructed by choosing specific functions $c(b^2)$, $\mu(b^2)$, $\nu(b^2)$, and $h(b^2)$?

Key findings

• The paper establishes a complete characterization of non-Riemannian general $(\alpha,\beta)$-metrics with vanishing Douglas curvature via a system of two equations: a second-order PDE in $\phi$ (Equation 1.3) and a condition on $b_{i|j}$ (Equation 1.4).
• A general solution to the PDE (1.3) is constructed by introducing a change of variables $\zeta(b^2, s)$, leading to $\phi(b^2,s) = h(b^2)s + \frac{(1+b^2)(1+b^4-b^2s^2)+s^2(1-b^2)}{(1+b^4)^2}\sqrt{\frac{(1-b^2)e^{b^2}}{1+b^4-b^2s^2}}$, where $h(b^2)$ is an arbitrary $C^\infty$ function.
• When $\alpha$ is projectively flat (constant curvature), a special solution is constructed with $\alpha = \frac{\sqrt{(1+\kappa|x|^2)|y|^2 - \kappa\langle x,y\rangle^2}}{1+\kappa|x|^2}$ and a specific $\beta$ involving constants $\delta_1, \delta_2$, and a vector $a$, satisfying the required $b_{i|j}$ condition.
• For $\alpha$ conformal to $|y|$, two explicit examples are given: one with $\alpha = \frac{1}{2|x|}|y|$, $\beta = 2\varepsilon e^{-|x|^2}\langle x,y\rangle$, and another with $\alpha = \frac{1}{1+|x|^2}|y|$, $\beta = \frac{1+|x|^2}{1-|x|^2}\langle x,y\rangle$, both yielding valid solutions to Equation (1.4).
• The solution structure allows for a wide class of new Douglas metrics by freely choosing $c(b^2)$, $\mu(b^2)$, $\nu(b^2)$, and $h(b^2)$, demonstrating the richness of the class beyond previously known cases.
• The results generalize prior work by removing the restrictive assumptions of $\beta$ being closed or conformal, thus providing a comprehensive framework for constructing non-Riemannian Douglas metrics.