  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 $\rho_k$ 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 $\rho_{k,k+1}$ of two
consecutive sites that is needed to evaluate the energy of two-body
Hamiltonians, and for the reduced density matrix $\rho_{k,k+r}$ 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.
