11 research outputs found
Groups of diffeomorphisms and geometric loops of manifolds over ultra-normed fields
The article is devoted to the investigation of groups of diffeomorphisms and
loops of manifolds over ultra-metric fields of zero and positive
characteristics. Different types of topologies are considered on groups of
loops and diffeomorphisms relative to which they are generalized Lie groups or
topological groups. Among such topologies pairwise incomparable are found as
well. Topological perfectness of the diffeomorphism group relative to certain
topologies is studied. There are proved theorems about projective limit
decompositions of these groups and their compactifications for compact
manifolds. Moreover, an existence of one-parameter local subgroups of
diffeomorphism groups is investigated.Comment: Some corrections excluding misprints in the article were mad
On a global differential geometric approach to the rational mechanics of deformable media
In the past the rational mechanics of deformable media was largely concerned with materials governed by linear constitutive equations. In recent years, the theory has expanded considerably towards covering materials for which the constitutive equations are inherently nonlinear, and/or whose mechanical properties resemble in some respects those of a fluid and in others those of a solid. In the present article we formulate a satisfactory global mathematical theory of moving deformable media, which includes all these aspects
Variational methods in shape analysis
The analysis of shapes as elements in a frequently infinite-dimensional space of shapes has attracted increasing attention over the last decade. There are pioneering contributions in the theoretical foundation of shape space as a Riemannian manifold as well as path-breaking applications to quantitative shape comparison, shape recognition, and shape statistics. The aim of this chapter is to adopt a primarily physical perspective on the space of shapes and to relate this to the prevailing geometric perspective. Indeed, we here consider shapes given as boundary contours of volumetric objects, which consist either of a viscous fluid or an elastic solid. In the first case, shapes are transformed into each other via viscous transport of fluid material, and the flow naturally generates a connecting path in the space of shapes. The viscous dissipation rateâthe rate at which energy is converted into heat due to frictionâcan be defined as a metric on an associated Riemannian manifold. Hence, via the computation of shortest transport paths one defines a distance measure between shapes