Journal of the
Korean Mathematical Society JKMS

In the present paper, we investigate the geometric structures of the Dirichlet manifold composed of the Dirichlet distribution. We show that the Dirichlet distribution is an exponential family distribution. We consider its dual structures and give its geometric metrics, and obtain the geometric structures of the lower dimension cases of the Dirichlet manifold. In particularly, the Beta distribution is a 2-dimensional Dirichlet distribution. Also, we construct an affine immersion of the Dirichlet manifold. At last, we give the $e$-flat hierarchical structures and the orthogonal foliations of the Dirichlet manifold. All these work will enrich the theoretical work of the Dirichlet distribution and will be great help for its further applications.

Keywords: Dirichlet manifold, $\alpha$-geometric structures, foliation, immersion

MSC numbers: 53B20, 62B10