Explicit properties of the simplest inhomogeneous Matrix-Product-State including the Riemann metric of the MPS manifold

We consider the simplest inhomogeneous Matrix-Product-State for an open chain of N quantum spins that involves only two angles per site and two angles per bond with the following direct physical meanings. The two angles associated to the site k are the two Bloch angles that parametrize the two orthonormal eigenvectors of the reduced density matrix ρ_k of the spin k alone. The two angles associated to the bond (k,k+1) parametrize the entanglement properties of the Schmidt decomposition across the bond (k,k+1). Explicit results are given for the reduced density matrix ρ_k,k+1 of two consecutive sites that is needed to evaluate the energy of two-body Hamiltonians, and for the reduced density matrix ρ_k,k+r of two sites at distance r that is needed to evaluate the spin-spin correlations at distance r. The global structure of the MPS manifold as parametrized by these (4N-2) angles is then characterized by its explicit Riemann metric. Finally, the generalizations to any tree-like structure without loops and to the chain with periodic boundary conditions are discussed.