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**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)

### MSC:

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |