  The present paper is devoted to the study of geometry of Batalin-Vilkovisky
quantization procedure. The main mathematical objects under consideration are
P-manifolds and SP-manifolds (supermanifolds provided with an odd symplectic
structure and, in the case of SP-manifolds, with a volume element). The
Batalin-Vilkovisky procedure leads to consideration of integrals of the
superharmonic functions over Lagrangian submanifolds. The choice of Lagrangian
submanifold can be interpreted as a choice of gauge condition; Batalin and
Vilkovisky proved that in some sense their procedure is gauge independent. We
prove much more general theorem of the same kind. This theorem leads to a
conjecture that one can modify the quantization procedure in such a way as to
avoid the use of the notion of Lagrangian submanifold. In the next paper we
will show that this is really so at least in the semiclassical approximation.
Namely the physical quantities can be expressed as integrals over some set of
critical points of solution S to the master equation with the integrand
expressed in terms of Reidemeister torsion. This leads to a simplification of
quantization procedure and to the possibility to get rigorous results also in
the infinite-dimensional case. The present paper contains also a compete
classification of P-manifolds and SP-manifolds. The classification is
interesting by itself, but in this paper it plays also a role of an important
tool in the proof of other results.
