Transport information geometry: Riemannian calculus on probability simplex

We formulate the Riemannian calculus of probability set embedded with $$L^2$$ -Wasserstein metric. This is an initial work transport information geometry. Our investigation starts simplex (probability manifold) supported on vertices a finite graph. The main idea to embed manifold as submanifold positive measure space weighted graph Laplacian operator. By this viewpoint, we establish torsion–free Christoffel symbols, Levi–Civita connections, curvature tensors and volume forms in by Euclidean coordinates. As consequence, Jacobi equation, Laplace-Beltrami, Hessian operators diffusion processes are derived. These geometric computations also provided infinite-dimensional density (density finite-dimensional manifold. In particular, present identity connecting among Baker–Émery $$\Gamma _2$$ operator (carré du champ itéré), Fisher–Rao metric optimal Several examples demonstrated.