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Extremal immersions and the extended frame bundle

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Publication date1991
Host publicationGeometry of low-dimensional manifolds: 1: gauge theory and algebraic surfaces
EditorsS. K. Donaldson, C. B. Thomas
Place of PublicationCambridge
PublisherCambridge University Press
Number of pages24
ISBN (Electronic)9780511629334
ISBN (Print)9780521399784
<mark>Original language</mark>English

Publication series

NameLondon Mathematical Society Lecture Note Series
ISSN (Print)0076-0552


We present a computationally powerful formulation of variational problems that depend on the extrinsic and intrinsic geometry of immersions into a manifold. The approach is based on a lift of the action integral to a larger space and proceeds by systematically constraining the variations to preserve the foliation of a Pfaffian system on an extended frame bundle. Explicit Euler-Lagrange equations are computed for a very general class of Lagrangians and the method illustrated with examples relevant to recent developments in theoretical physics. The method provides a means of determining spatial boundary conditions for immersions with boundary and enables a construction to be made of constants of the motion in terms of Euler- Lagrange solutions and admissible symmetry vectors.