11,834 research outputs found
Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy
Nurowski showed that any generic 2-plane field on a 5-manifold
determines a natural conformal structure on ; these conformal
structures are exactly those (on oriented ) whose normal conformal holonomy
is contained in the (split, real) simple Lie group . Graham and Willse
showed that for real-analytic the same holds for the holonomy of the
real-analytic Fefferman-Graham ambient metric of , and that both holonomy
groups are equal to for almost all . We investigate here independently
interesting plane fields for which the associated holonomy groups are a proper
subset of .
Cartan solved the local equivalence problem for -plane fields and
constructed the fundamental curvature tensor for these objects. He
furthermore claimed to describe locally all whose infinitesimal symmetry
algebra has rank at least and gave a local quasi-normal form, depending on
a single function of one variable, for those that furthermore satisfy a natural
degeneracy condition on , but Doubrov and Govorov recently rediscovered a
counterexample to Cartan's claim. We show that for all given by Cartan's
alleged quasi-normal form, the conformal structures induced via
Nurowski's construction are almost Einstein, that we can write their ambient
metrics explicitly, and that the holonomy groups associated to are always
the -dimensional Heisenberg group, which here acts indecomposably but not
irreducibly. (Not all of these properties hold, however, for Doubrov and
Govorov's counterexample.) We also show that the similar results hold for the
related class of -plane fields defined on suitable jet spaces by ordinary
differential equations satisfying a simple genericity
condition.Comment: 34 pages. Revised to accommodate a counterexample to a cited
classification of Cartan found by Doubrov and Govorov; fixed some minor
error
On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
We show how to find a complete set of necessary and sufficient conditions
that solve the fixed-parameter local congruence problem of immersions in
-spaces, whether homogeneous or not, provided that a certain
order jet bundle over the -space admits a -invariant local coframe field
of constant structure. As a corollary, we note that the differential order of a
minimal complete set of congruence invariants is bounded by . We
demonstrate the method by rediscovering the speed and curvature invariants of
Euclidean planar curves, the Schwarzian derivative of holomorphic immersions in
the complex projective line, and equivalents of the first and second
fundamental forms of surfaces in subject to rotation
An introduction to Lie group integrators -- basics, new developments and applications
We give a short and elementary introduction to Lie group methods. A selection
of applications of Lie group integrators are discussed. Finally, a family of
symplectic integrators on cotangent bundles of Lie groups is presented and the
notion of discrete gradient methods is generalised to Lie groups
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