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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)
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 1-parametrization of the roots exists. Unfortunately there was an error in the calculation of b 3 and we have been informed by(More)
Smooth, real analytic and holomorphic mappings deened on non-open subsets of innnite dimensional vector spaces are treated. 0. Introduction In this paper we will generalize the concept of diierentiable maps f : E X ! F deened on open subsets to such on more general subsets of innnite dimensional vector spaces. We will refer to the theories for open domains(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)
If A(t) is a C 1,α-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be pa-rameterized C 1 in t. If A is C ∞ then the eigenvalues can be parameterized twice differentiable. Theorem. Let t → A(t) for t ∈ R be a curve of unbounded self-adjoint operators in a Hilbert space with common(More)