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

SO(2N)/U(N) Riccati-Hartree-Bogoliubov Equation Based on the SO(2N) Lie Algebra of the Fermion Operators