In this article we attempt to formulate Riemannian and Randers-Finsler metrics in information geometry and study their mechanical properties. Starting from the gradient flow equations, we show how to formulate Riemannian metrics, and demonstrate their duality under canonical transformation. Then we show how to formulate a Randers-Finsler metric from deformed gradient equations. The theories described are applied to the Gaussian model and tested to verify consistency and also to Reissner-Nordstrom and Kerr black holes.

翻译：本文尝试在信息几何中构建黎曼度规和Randers-Finsler度规，并研究其力学性质。从梯度流方程出发，我们展示了如何构建黎曼度规，并证明了其在正则变换下的对偶性。随后又展示了如何从变形梯度方程构建Randers-Finsler度规。所描述的理论被应用于高斯模型以验证其一致性，同时也应用于Reissner-Nordstrom黑洞和Kerr黑洞。