Let Ω = (ω1,…,ω
) be differential 1-forms with polynomial coefficients in
. A Pfaffian manifold of Ω is by definition a maximal integral k-manifold of Ω. It is shown that the number of homeomorphism classes of all Pfaffian manifolds of Rolle Type of Ω is finite and, moreover, bounded by a computable function in variables n, k and the degree of ω
. Finiteness is proved also in any o-minimal structure.
We give also an example of a semi-algebraic C
1 differential form on a semi-algebraic C2
3-manifold whose Pfaffian manifolds have homeomorphism classes of the cardinality of continuum. Hence the cardinality of all manifolds is the continuum (not countable).