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| |**Abstract**: Homogeneous spaces are spaces with built-in symmetry, such as planes, spheres, or hyperbolic spaces. We consider numerical integrators of ODEs on homogeneous spaces. One obtain such integrators by combining a Lie group integrator with an isotropy map. In order for the integrators to preserve the intrinsic symmetry of the homogeneous space, we show that the isotropy map must be equivariant. In order to show that, we introduce a novel, all encompassing description of Lie group integrators, comprising most of the existing Lie group methods. These equivariant isotropy maps can be identified with invariant connections. Such connections may or may not exist, and can be given either implicitly or explicitly. Examples include Lie groups, affine spaces, Stiefel, Grassmann, isospectral manifolds, and fixed rank matrices. Finally, one can consider higher order isotropy choices. In the affine case, they have in fact highest order one, and form an affine space of dimension two. These isotropy choices are related to exponential integrators. | | |**Abstract**: Homogeneous spaces are spaces with built-in symmetry, such as planes, spheres, or hyperbolic spaces. We consider numerical integrators of ODEs on homogeneous spaces. One obtain such integrators by combining a Lie group integrator with an isotropy map. In order for the integrators to preserve the intrinsic symmetry of the homogeneous space, we show that the isotropy map must be equivariant. In order to show that, we introduce a novel, all encompassing description of Lie group integrators, comprising most of the existing Lie group methods. These equivariant isotropy maps can be identified with invariant connections. Such connections may or may not exist, and can be given either implicitly or explicitly. Examples include Lie groups, affine spaces, Stiefel, Grassmann, isospectral manifolds, and fixed rank matrices. Finally, one can consider higher order isotropy choices. In the affine case, they have in fact highest order one, and form an affine space of dimension two. These isotropy choices are related to exponential integrators. | |