  In this paper we present the induced representation of SO(2N) canonical
transformation group and introduce SO(2N)/U(N) coset variables. We give a
derivation of the time dependent Hartree-Bogoliubov (TDHB) equation on the
Kaehler coset space G/H=SO(2N)/U(N) from the Euler-Lagrange equation of motion
for the coset variables. The TDHB wave function represents the TD behavior of
Bose condensate of fermion pairs. It is a good approximation for the ground
state of the fermion system with a pairing interaction, producing the
spontaneous Bose condensation. To describe the classical motion on the coset
manifold, we start from the local equation of motion. This equation becomes a
Riccati-type equation. After giving a simple two-level model and a solution for
a coset variable, we can get successfully a general solution of TDRHB equation
for the coset variables. We obtain the Harish-Chandra decomposition for the
SO(2N) matrix based on the nonlinear Moebius transformation together with the
geodesic flow on the manifold.
