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Interactions between para-quaternionic and Grassmannian geometry


Almost para-quaternionic structures on smooth manifolds of dimension 2n are equivalent to almost Grassmannian structures of type (2, n). We remind the equivalence and exhibit some interrelations between subjects that were previously studied independently from the para-quaternionic and the Grassmannian point of view. In particular, we relate the respective normalization conditions, distinguished curves, and twistor constructions.

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  1. Instead of the prefix para-, various synonyms can be found in the literature; split- is probably the most often.

  2. Throughout the paper, we use the standard notation of favorite Lie groups; GL for general linear, PGL for projective linear, SL for special linear, SO for special orthogonal, Spin for spin. The corresponding Lie algebras will be denoted as \(\mathfrak {gl}\), \(\mathfrak {sl}\) and \(\mathfrak {so}\), respectively.

  3. Almost para-quaternionic structures appear under various names in the literature; the frequent one in older references is almost quaternionic structures of second type. There are also alternative equivalent definitions of the structure; see, e.g., [22] for more information.

  4. Be aware that the conventions in references are not always consistent; we follow the one in which the \(\beta \)-integrability corresponds to the anti-self-duality.

  5. Beside an enormous terminology related to concrete geometries, these curves have various general nicknames, e.g., Cartan’s circles, generalized geodesics or canonical curves.

  6. Comparing with the original formulation, we differ in the sign in the front of the term containing \({\text{ P }}\). This just reflects the difference in the definition of Rho tensor here, which we took from [9], and in [7].


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I am grateful to Dmitri V. Alekseevsky, Andreas Čap, Jan Slovák and Josef Šilhan for many helpful conversations.

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Correspondence to Vojtěch Žádník.

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This work was supported by the Czech Science Foundation (GAČR) under the Grant GA17-01171S.

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Žádník, V. Interactions between para-quaternionic and Grassmannian geometry. Ann Glob Anal Geom 57, 321–347 (2020).

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  • Almost para-quaternionic structures
  • Almost Grassmannian structures
  • Cartan connections

Mathematics Subject Classification

  • 53C15
  • 53C05