Andreas Kriegl

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The aim of this book is to lay foundations of differential calculus in infinite dimensions and to discuss those applications in infinite dimensional differential geometry and global analysis which do not involve Sobolev completions and fixed point theory. The approach is very simple: A mapping is called smooth if it maps smooth curves to smooth curves. All(More)
We investigate the rudiments of Riemannian geometry on orbit spaces M/G for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space M/G and they can hit strata which are more singular only at the end points. This is phrased as convexity result. The geodesic spray, viewed as a(More)
Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an ‘evolution operator’ exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal(More)
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Convenient vector spaces . . . . . . . . . . . . . . . . . . . . . . 4 2. Non-commutative differential forms . . . . . . . . . . . . . . . . . 7 3. Some related questions . . . . . . . . . . . . . . . . . . . . . . . 14 4. The calculus of Frölicher and Nijenhuis . . . . . . . . . .(More)
For Denjoy–Carleman differential function classes C where the weight sequence M = (Mk) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is C if it maps C -curves to C -curves. The category of C -mappings is cartesian closed in the sense that C (E, C (F, G)) = C (E × F, G) for(More)
We show that the roots of any smooth curve of polynomials with real roots only can be parametrized twice differentiable (but not better). In [1] we claimed that there exists a smooth curve of polynomials of degree 3 for which no C-parametrization of the roots exists. Unfortunately there was an error in the calculation of b3 and we have been informed by(More)