  Extended supersymmetric sigma-model is given, standing on the SO(2N+1) Lie
algebra of fermion operators composed of annihilation-creation operators and
pair operators. Canonical transformation, the extension of the SO(2N)
Bogoliubov transformation to the SO(2N+1) group, is introduced. Embedding the
SO(2N+1) group into an SO(2N+2) group and using SO(2N+2)/U(N+1) coset
variables, we investigate a new aspect of the supersymmetric sigma-model on the
Kaehler manifold of the symmetric space SO(2N+2)/U(N+1). We construct a Killing
potential which is just the extension of the Killing potential in the
SO(2N)/U(N) coset space given by van Holten et al. to that in the
SO(2N+2)/U(N+1) coset space. To our great surprise, the Killing potential is
equivalent with the generalized density matrix. Its diagonal-block matrix is
related to a reduced scalar potential with a Fayet-Ilipoulos term. The reduced
scalar potential is optimized in order to see the behaviour of the vacuum
expectation value of the sigma-model fields and a proper solution for one of
the SO(2N+1) group parameters is obtained. We give bosonization of the SO(2N+2)
Lie operators, vacuum functions and differential forms for their bosons
expressed in terms of the SO(2N+2)/U(N+1) coset variables, a U(1) phase and the
corresponding Kaehler potential.
