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Integrators on Homogeneous Spaces: Isotropy Choice and Connections

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Abstract

We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an isotropy choice. We show that equivariant isotropy choices combined with equivariant Lie group integrators produce equivariant homogeneous space integrators. Moreover, we show that the RKMK, Crouch–Grossman, or commutator-free methods are equivariant. To show this, we give a novel description of Lie group integrators in terms of stage trees and motion maps, which unifies the known Lie group integrators. We then proceed to study the equivariant isotropy maps of order zero, which we call connections, and show that they can be identified with reductive structures and invariant principal connections. We give concrete formulas for connections in standard homogeneous spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar decomposition manifolds. Finally, we show that the space of matrices of fixed rank possesses no connection.

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Notes

  1. With the notation (21) that isotropy choice is simply \(\langle \varpi ,f \rangle = f\) hence the name of tautological isotropy map.

  2. The interested reader is referred to [22], or [15, § X.2].

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Acknowledgments

This research was supported by the Spade Ace Project and by the J. C. Kempe memorial fund.

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Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Munthe-Kaas

  2. Department of Mathematics and Statistics, Umeå University, Umeå, Sweden

    Olivier Verdier

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  1. Hans Munthe-Kaas
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  2. Olivier Verdier
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Correspondence to Olivier Verdier.

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Communicated by Ernst Hairer.

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Munthe-Kaas, H., Verdier, O. Integrators on Homogeneous Spaces: Isotropy Choice and Connections. Found Comput Math 16, 899–939 (2016). https://doi.org/10.1007/s10208-015-9267-7

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  • DOI: https://doi.org/10.1007/s10208-015-9267-7

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