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Intrinsic holonomy and curved cosets of Cartan geometries

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Abstract

We provide an intrinsic notion of curved cosets for arbitrary Cartan geometries, simplifying the existing construction of curved orbits for a given holonomy reduction. To do this, we intrinsically define a holonomy group, which is shown to coincide precisely with the standard definition of the holonomy group for Cartan geometries in terms of an associated principal connection. These curved cosets retain many characteristics of their homogeneous counterparts, and they behave well under the action of automorphisms. We conclude the paper by using the machinery developed to generalize the de Rham decomposition theorem for Riemannian manifolds and give a potentially useful characterization of inessential automorphism groups for parabolic geometries.

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Notes

  1. To clarify, throughout the paper we use the terms “intrinsic” and “natural” to mean that we are only using concepts native to Cartan geometries, as opposed to the more general machinery of principal connections. We are not disputing that the holonomy group defined using the extended principal bundle nor the results from [8] are canonical to the underlying Cartan geometries.

  2. See, for example, Proposition 2.1.1 of [5], which notes that the subgroups corresponding to holonomy from contractible loops coincide.

  3. We say that a Cartan geometry \((\mathscr {G},\omega )\) of type (G, H) is reductive if and only if there exists an \(\mathrm {Ad}_H\)-invariant subspace \(\mathfrak {m}\subseteq \mathfrak {g}\) such that \(\mathfrak {g}\approx \mathfrak {m}{\oplus }\mathfrak {h}\) as an \(\mathrm {Ad}_H\)-module.

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Acknowledgements

The author would like to thank Andreas Čap for his helpful commentary on several drafts of this paper, in particular for his critique of earlier drafts.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Jacob W. Erickson

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Correspondence to Jacob W. Erickson.

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Erickson, J.W. Intrinsic holonomy and curved cosets of Cartan geometries. European Journal of Mathematics 8, 446–474 (2022). https://doi.org/10.1007/s40879-022-00535-7

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