  We formulate the Riemannian calculus of the probability set embedded with
$L^2$-Wasserstein metric. This is an initial work of transport information
geometry. Our investigation starts with the probability simplex (probability
manifold) supported on vertices of a finite graph. The main idea is to embed
the probability manifold as a submanifold of the positive measure space with a
nonlinear metric tensor. Here the nonlinearity comes from the linear weighted
Laplacian operator. By this viewpoint, we establish torsion-free Christoffel
symbols, Levi-Civita connections, curvature tensors and volume forms in the
probability manifold by Euclidean coordinates. As a consequence, the Jacobi
equation, Laplace-Beltrami and Hessian operators on the probability manifold
are derived. These geometric computations are also provided in the
infinite-dimensional density space (density manifold) supported on a
finite-dimensional manifold. In particular, an identity is given connecting the
Baker-{\'E}mery $\Gamma_2$ operator (carr{\'e} du champ it{\'e}r{\'e}) by
connecting Fisher-Rao information metric and optimal transport metric. Several
examples are demonstrated.
