A survey on non-archimedean immersions (Q2773750)

This paper is a detailed survey on non-archimedean analysis, including a selection of the definitions, notions and results published during the last thirty years which are used, as important instruments of study, in current research of local and global analysis on manifolds, and also in many other fields of Mathematics and Physics.NEWLINENEWLINENEWLINEMore precisely, the author recalls the definitions of ultrametric normed field, norm on an ordered field, completion of a topological non-archimedean valued field, spherical completeness of a non-archimedean metric space, non-archimedean Banach space, differentiation in a non-archimedean Banach space and immersion of a differentiable manifold in an ultrametric Banach space. Furthermore, he states some original results about structure equations, ultrametric volume formula on a typical subbundle and absolute curvature of a \(p\)-adic manifold immersed in a finite dimensional Banach space.NEWLINENEWLINENEWLINEBesides the definitions and most important examples, the author quotes some well-known theorems characterizing the above-mentioned notions, and gives necessary and/or sufficient conditions. Although the proofs of the stated theorems are omitted, because of the limitation in length, this does not weaken the overall impression, since it is clearly pointed out where the proofs can be found in the indicated references.NEWLINENEWLINENEWLINEMoreover, in all instances the author emphasizes the differences between non-archimedean analysis and real and complex analysis. He also indicates those results that could be generalized and others that could not, accompanied by detailed explanations.