  The geometrical description of a Hilbert space asociated with a quantum
system considers a Hermitian tensor to describe the scalar inner product of
vectors which are now described by vector fields. The real part of this tensor
represents a flat Riemannian metric tensor while the imaginary part represents
a symplectic two-form. The immersion of classical manifolds in the complex
projective space associated with the Hilbert space allows to pull-back tensor
fields related to previous ones, via the immersion map. This makes available,
on these selected manifolds of states, methods of usual Riemannian and
symplectic geometry. Here we consider these pulled-back tensor fields when the
immersed submanifold contains separable states or entangled states. Geometrical
tensors are shown to encode some properties of these states. These results are
not unrelated with criteria already available in the literature. We explicitly
deal with some of these relations.
