Abstract
Following Rao's idea to use the Fisher information matrix (FIM) as a Riemannian metric, we show that the family of Gumbel distributions determines a two dimensional Riemannian manifold. In this paper we illustrates the information geometry of the Gumbel space, and derive he geometry as; connections, curvature tensor, Ricci curvature with its eigenvalues and eigenvectors, scalar curvature, sectional curvature and mean curvature, where we show that Gumbel space has a negative constant scalar curvature. Moreover, we prove that log-Gumbel manifold is an isometric isomorphic of the origin manifold, which is important in stochastic process since Gumbel distributions are related to exponential distributions.