19 research outputs found
Isometric Immersions of Space Forms and Soliton Theory
This paper studies isometric immersions of space forms by means of a
hierarchy of finite dimensional integrable systems in Lax form on loop
algebras.Comment: 15 pages, latex2e, postscript available at
http://www_sfb288.math.tu-berlin.de/preprints.htm
Curved flats in symmetric spaces
In this paper we study maps (curved flats) into symmetric spaces which are
tangent at each point to a flat of the symmetric space. Important examples of
such maps arise from isometric immersions of space forms into space forms via
their Gauss maps. Further examples are found in conformal geometry, e.g. the
curved flats obtained from isothermic surfaces and conformally flat 3-folds in
the 4-sphere. Curved flats admit a 1-parameter family of deformations (spectral
parameter) which enables us to make contact to integrable system theory. In
fact, we give a recipe to construct curved flats (and thus the above mentioned
geometric objects) from a hierarchy of finite dimensional algebraically
completely integrable flows.Comment: 9 pages, latex2e, no figures, also available at
http://www_sfb288.math.tu-berlin.de/preprints.htm
Progress in the Theory of Singular Riemannian Foliations
A singular foliation is called a singular Riemannian foliation (SRF) if every
geodesic that is perpendicular to one leaf is perpendicular to every leaf it
meets. A typical example is the partition of a complete Riemannian manifold
into orbits of an isometric action.
In this survey, we provide an introduction to the theory of SRFs, leading
from the foundations to recent developments in research on this subject.
Sketches of proofs are included and useful techniques are emphasized. We study
the local structure of SRFs in general and under curvature conditions. We
review the solution of the Palais-Terng problem on integrability of the
horizontal distribution. Important special classes of SRFs, like polar and
variationally complete foliations and their connections, are treated. A
characterisation of SRFs whose leaf space is an orbifold is given. Moreover,
desingularizations of SRFs are studied and applications, e.g., to Molino's
conjecture, are presented