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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)

- Andreas Kriegl, Mark Losik, Peter W. Mi
- 1998

We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given. Table of contents

- Dmitri Alekseevsky, Andreas Kriegl, Mark Losik, Peter W. Michor, Peter W. Michor
- 2003

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)

Let C[M ] be a (local) Denjoy–Carleman class of Beurling or Roumieu type, where the weight sequence M = (Mk) is log-convex and has moderate growth. We prove that the groups DiffB[M ](Rn), DiffW [M ],p(Rn), DiffS [M ] [L] (Rn), and DiffD[M ](Rn) of C[M ]-diffeomorphisms on Rn which differ from the identity by a mapping in B[M ] (global Denjoy– Carleman), W… (More)

- Dmitri Alekseevsky, Andreas Kriegl, Mark Losik, Peter W. Michor
- 1997

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.

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)

We present here ”the” cartesian closed theory for real analytic mappings. It is based on the concept of real analytic curves in locally convex vector spaces. A mapping is real analytic, if it maps smooth curves to smooth curves and real analytic curves to real analytic curves. Under mild completeness conditions the second requirement can be replaced by:… (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)

- Andreas Kriegl, Mark Losik
- 2002

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)