DIFFERENTIAL GEOMETRY OF GRASSMANN MANIFOLDS

The Geometry and Topology on Grassmann Manifolds

This paper shows that the Grassmann Manifolds GF(n,N) can all be imbedded in an Euclidean space MF(N) naturally and the imbedding can be realized by the eigenfunctions of Laplacian △ on GF(n,N). They are all minimal submanifolds in some spheres of MF(N) respectively. Using these imbeddings, we construct some degenerate Morse functions on Grassmann Manifolds, show that the homology of the comple...

Methods of Modern Differential Geometry in Quantum Chemistry: TD Theories on Grassmann and Hartree-Fock Manifolds

ABSTRACT: Hamiltonian and Schrödinger evolution equations on finitedimensional projective space are analyzed in detail. Hartree-Fock (HF) manifold is introduced as a submanifold of many electron projective space of states. Evolution equations, exact and linearized, on this manifold are studied. Comparison of matrices of linearized Schrödinger equations on many electron projective space and on t...

Differential Calculus on Quantum Complex Grassmann Manifolds I: Construction

Covariant first order differential calculus over quantum complex Grassmann manifolds is considered. It is shown by a Pusz–Woronowicz type argument that under restriction to calculi close to classical Kähler differentials there exist exactly two such calculi for the homogeneous coordinate ring. Complexification and localization procedures are used to induce covariant first order differential cal...

Space Forms of Grassmann Manifolds

Cobordism independence of Grassmann manifolds

This paper is a continuation of the ongoing study of cobordism of Grassmann manifolds. Let F denote one of the division rings R of reals, C of complex numbers, or H of quaternions. Let t = dimRF . Then the Grassmannian manifold Gk(F) is defined to be the set of all k-dimensional (left) subspaces of Fn+k. Gk(F) is a closed manifold of real dimension nkt. Using the orthogonal complement of a subs...