Differential calculus over general base fields and rings

Some aspects of differential theories Respectfully dedicated to the memory of Serge Lang

“…As a consequence, there are several theories of infinite-dimensional manifolds, Lie groups and differential geometric structures. Changing the real or complex ground field to a more general topological field or ring, even more general differential calculus, Lie theory and differential geometry may be constructed [3,4]. In this subsection we briefly explain the approach to differential calculus originated by A. D. Michel [25] and A. Bastiani [2], and popularized by J. Milnor [27] and R. Hamilton [13].…”

Every smooth p-adic Lie group admits a compatible analytic structure

“…Then, various (new) lemmas are formulated, most of which are based on Taylor expansions and generalize classical facts from multivariable calculus over R. We recommend to take these lemmas on faith at this point; if desired, the proofs can be looked up later in Appendix A. Definition 1.1 Let K be a non-discrete (Hausdorff) topological field, E and F be (Hausdorff) topological K-vector spaces, and f : U → F be a map on an open subset of E. We call f C 0 if it is continuous; it is C 1 if it is C 0 and if there exists a (necessarily unique) continuous map f [1] :…”

Projective Completions of Jordan Pairs, Part II: Manifold Structures and Symmetric Spaces

“…(8) There are several versions of Taylor's formula (see [BGN03]), but none of them will be used in this work.…”

“…[BGN03]); in particular, there is a natural manifold structure (with atlas T A) on T M such that the natural projection π : T M → M is smooth; the tangent space T p M is defined to be the fiber π −1 (p). If f : M → N is C k , there is a well-defined tangent map T f : T M → T N , and we have the usual functorial properties (including compatibility with direct products:…”

“…Compared with the approach from [Up85], our approach is more algebraic and less analytic, which makes it considerably simpler and more elementary. The algebraic results from Part I of this work ( [BN03]) which we need are summarized in Chapter 4, and the basic notions of differential calculus and manifolds over general topological fields and rings from [BGN03] are recalled in Chapter 1. The reader who is only interested in the real or complex theory may everywhere replace K by R or C , and he will see that all notions from calculus we use are the ones which he is used to.…”