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In Bourbaki "Variétés différentielles et analytiques" there is a statement (without proof) that for a vector field on smooth manifold over complete normed field (characteristic zero) there is a local unique flow along it. Is it true or not? Is there any proof for this statement?

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No idea. What are the basic definitions here for smooth manifold and vector field over one of your complete normed fields? Over the reals, there is no difference between defining a vector as a derivation of functions at a point or as an equivalence class of curves going through the same point. –  Will Jagy Sep 6 '11 at 20:18
    
I'm trying to get rid of the ambiguous [differential-equations] tag, usually in favor of [ca.analysis-and-odes] or [ap.analysis-of-pdes] but neither alternative seems to apply in this case. If you have an idea for a substitute that applies to this case, I will gladly create it for you. –  François G. Dorais Sep 7 '11 at 2:19
    
@Will: you know the definition of smooth functions, you can check the Implicit function theorem for complete normed fields (proof in Rudin's book on calculus is good for it), so you have a definition of smooth manifold. Over p-adic numbers (and Witt vectors over $\mathbb F_{p^n}$) there is no difference betweeb derivations and vector fields too. Maybe someone know the proof for p-adic numbers? I didn't find any such a statement (and the contrary too) in textbooks (like the Dwork's one) on p-adic differential equations. –  zroslav Sep 7 '11 at 9:02
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zroslav: maybe you should be less quick to assume everyone knows what a smooth function is over a complete normed field besides R. Although I don't have Bourbaki's book in front of me I would guess that in it what's defined geometrically over complete normed fields besides R is not the concept of a smooth manifold using infinitely differentiable functions, but rather an analytic manifold using functions that are given by power series. –  KConrad Sep 7 '11 at 11:48
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In fact, there is a notion of smoothness for the p-adic case introduced by Schikhof in his book "Ultrametric Calculus", and there are papers by Ludkovsky (Lyudkovskij) who develops a theory of manifolds based on Schikhof's definition. However I did not study those papers in detail. –  Anatoly Kochubei Sep 8 '11 at 6:06

1 Answer 1

In the p-adic case, the standard theory deals not with smooth functions and manifolds but with analytic ones. A complete exposition including necessary results from analysis is given by J.-P. Serre, "Lie algebras and Lie groups", New York, Benjamin, 1965.

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Yes, the analytic case is rather simpler –  zroslav Sep 8 '11 at 7:01

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