  The integrability-condition method is regarded as a mathematical tool to
describe the symmetry of collective sub-manifold. We here adopt the
particle-hole representation. In the conventional time-dependent (TD)
self-consistent field (SCF) theory, we take the one-form linearly composed of
the TD SCF Hamiltonian and the infinitesimal generator induced by the
collective-variable differential of canonical transformation on a group.
Standing on the differential geometrical viewpoint, we introduce a
Lagrange-like manner familiar to fluid dynamics to describe collective
coordinate systems. We construct a geometric equation, noticing the structure
of coset space SU(m+n)/S(U(m) x U(n)). To develop a perturbative method with
the use of the collective variables, we aim at constructing a new fermion SCF
theory, i.e., renewal of TD Hartree-Fock (TDHF) theory by using the canonicity
condition under the existence of invariant subspace in the whole HF space. This
is due to a natural consequence of the maximally decoupled theory because there
exists an invariant subspace, if the invariance principle of Schredinger
equation is realized. The integrability condition of the TDHF equation
determining a collective sub-manifold is studied, standing again on the
differential geometric viewpoint. A geometric equation works well over a wide
range of physics beyond the random phase approximation.
