Calculus of flows on convenient manifolds. (English) Zbl 0881.58012

The author deals with the exponential mapping for the group of diffeomorphisms and with other mappings of similar character. He starts with a finite-dimensional smooth and compact manifold \(M\). The Lie algebra of the group of diffeomorphisms \(\text{Diff}(M)\) of \(M\) can be identified with the Lie algebra \(\Gamma (TM)\) of vector fields on \(M\). For \(X,Y\in \Gamma (TM)\) the \(X\)-derivative at \(tX\) in direction \(Y\) is defined to be \(D\exp(tX)Y=(d/ds)_0\exp t(X+sY)\). He computes first a formula for this \(X\)-derivative. Then he considers the special case of the exponential mapping (\(t=1\)) and presents several results concerning the injectivity, resp. the kernel, resp. the surjectivity of the exponential mapping. Next the author considers 1-parameter systems of diffeomorphisms, especially evolution flows, flows induced by the action of a regular Lie group, and 1-parameter families of bounded linear operators on a convenient vector space. Among other results, his main aim is to derive formulas analogous to the formula for the \(X\)-derivative. He uses the technique of the Frölicher-Kriegl differential calculus. The paper is very clearly written.
Reviewer: J.Vanžura (Brno)


58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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