In [1] and [2] we defined new invariants of both bipartite and multipartite quantum systems under local unitary transformations via algebraic-geometry of determinantal varieties, and gave a new separability criterion based on our new invariants. The purpose of this note is to show how Schmidt number of pure states in bipartite systems, a classical invariant, can be read out from algebraic-geome...

Pseudo Ricci symmetric real hypersurfaces of a complex projective space are classified and it is proved that there are no pseudo Ricci symmetric real hypersurfaces of the complex projective space CPn for which the vector field ξ from the almost contact metric structure (φ, ξ, η, g) is a principal curvature vector field.

An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended ∗email address: [email protected] to complex manifolds. Examples include the quantum sphere, the complex quantum projective spaces and the two-sheeted space.

It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions. 1. Projective connections. Let M be a complex manifold. Consider the following equivalence relation on the set of affine torsion-free connections on M : two connections Γ̂ and Γ are s...