Model-theoretic Differential Geometry

David A. Pierce

A differential field is here a structure (k, ∂_{0}, …, ∂_{m−1}), where k is a field of characteristic zero, and the unary functions ∂_{i} are commuting derivations. The model-completion of the theory of differential fields is axiomatized by the Frobenius Theorem of differential geometry. Similar arguments go through formally in case the derivations span a non-commutative Lie algebra.

The Frobenius Theorem is usually stated in terms of a real manifold, but it has a purely algebraic content. If K is a field-extension of k, then the possible extensions of the derivations ∂_{i} to K span a vector space, whose dual may be called the space of differential 1-forms of K over k. There are 1-forms dT^{i}, dual to the derivations ∂_{i}. A system of 1-forms ω of K has a generic solution if and only if the 2-forms dω are in the space generated by the forms ω, and the forms dT^{i} are linearly disjoint from the forms ω. (That the system has a generic solution means that the kernel of the forms ω is spanned by particular commuting extensions of the derivations ∂_{i})

Tracey McGrail (‘The model theory of partial differential fields’, J. Symbolic Logic, to appear) and Yoav Yaffe (‘Model completion of Lie differential fields’, Master’s Thesis, Hebrew University, 1998) have shown independently that the theory of differential fields has a model-completion, so their axioms must be equivalent to the Frobenius Theorem; but the latter can also be translated directly into an infinite set of first-order axioms.

Address: McMaster University, Department of Mathematics and Statistics, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4K1

E-mail: pierced@math.mcmaster.ca