Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields

We present an axiomatic approach to finiteand infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces over non-discrete topological fields, in particular ultrametric fields or the fields of real and complex numbers. In the latter case, a theory of differentiable mappings between general, not necessarily locally convex spaces is obtained, which in the locally convex case is equivalent to Keller’s C c -theory. AMS subject classification: 58C20, 22E65, 26E30, 26E15, 26E20, 46G05, 46G20, 46T25, 46A16, 46S10, 58A05, 39A70, 46T05, 58D05, 22E67.