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  • Print publication year: 1991
  • Online publication date: February 2010

13 - Extremal immersions and the extended frame bundle



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.


Current trends in theoretical physics have focussed attention on the properties of spacetime immersions that extremalise various aspects of their geometrical structure. Thus string theories are based on models that extremalise the induced area of two dimensional time-like world sheets. Their generalisations to p-dimensional immersions provide a dynamical prescription for (p – 1)-dimensional membranes. Extremalising the integral of the natural induced measure has provided a very rich phenomenological interpretation in the context of particle physics and has led to a number of speculations connecting gravitation to the other forces of Nature. In these developments the properties of the ambient embedding space for the various immersions play a minor role at the classical level. At the quantum level consistency conditions constrain their dimensionality when the ambient space is flat.

A number of recent papers [1] have begun to investigate the properties of spacetime immersions that extremalise integrals of certain of their extrinsic properties.