Page 1

Displaying 1 – 12 of 12

Showing per page

Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric

Claudio Altafini (2004)

ESAIM: Control, Optimisation and Calculus of Variations

For a riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the riemannian exponential...

Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric

Claudio Altafini (2010)

ESAIM: Control, Optimisation and Calculus of Variations

For a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the Riemannian exponential...

Representation of a gauge group as motions of a Hilbert space

Clara Lucía Aldana Domínguez (2004)

Annales mathématiques Blaise Pascal

This is a survey article based on the author’s Master thesis on affine representations of a gauge group. Most of the results presented here are well-known to differential geometers and physicists familiar with gauge theory. However, we hope this short systematic presentation offers a useful self-contained introduction to the subject.In the first part we present the construction of the group of motions of a Hilbert space and we explain the way in which it can be considered as a Lie group. The second...

Riemannian geometries on spaces of plane curves

Peter W. Michor, David Mumford (2006)

Journal of the European Mathematical Society

We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S 1 to the plane modulo the group of diffeomorphisms of S 1 , acting as reparametrizations. In particular we investigate the metric, for a constant A > 0 , G c A ( h , k ) : = S 1 ( 1 + A κ c ( θ ) 2 ) h ( θ ) , k ( θ ) | c ' ( θ ) | d θ where κ c is the curvature of the curve c and h , k are normal vector fields to c . The term A κ 2 is a sort of geometric Tikhonov regularization because, for A = 0 , the geodesic distance between any two distinct curves is 0, while for A > 0 the...

Currently displaying 1 – 12 of 12

Page 1