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DUALITY OF (2, 3, 5)-DISTRIBUTIONS AND LAGRANGIAN CONE STRUCTURES

Part of: Global differential geometry Classical differential geometry General theory of differentiable manifolds

Published online by Cambridge University Press:  23 January 2020

GOO ISHIKAWA Open the ORCID record for GOO ISHIKAWA [Opens in a new window] ,
YUMIKO KITAGAWA Open the ORCID record for YUMIKO KITAGAWA [Opens in a new window] ,
ASAHI TSUCHIDA Open the ORCID record for ASAHI TSUCHIDA [Opens in a new window]  and
WATARU YUKUNO Open the ORCID record for WATARU YUKUNO [Opens in a new window]
GOO ISHIKAWA
Affiliation:
Hokkaido University, Sapporo060-0810, Japan email ishikawa@math.sci.hokudai.ac.jp
YUMIKO KITAGAWA
Affiliation:
Oita National College of Technology, Oita870-0152, Japan email kitagawa@oita-ct.ac.jp
ASAHI TSUCHIDA
Affiliation:
Institute of Mathematics, Polish Academy of Science, Warszawa00-656, Poland email atsuchida@impan.pl
WATARU YUKUNO
Affiliation:
Hokkaido University, Sapporo060-0810, Japan email yukuwata@math.sci.hokudai.ac.jp
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Abstract

As was shown by a part of the authors, for a given $(2,3,5)$-distribution $D$ on a five-dimensional manifold $Y$, there is, locally, a Lagrangian cone structure $C$ on another five-dimensional manifold $X$ which consists of abnormal or singular paths of $(Y,D)$. We give a characterization of the class of Lagrangian cone structures corresponding to $(2,3,5)$-distributions. Thus, we complete the duality between $(2,3,5)$-distributions and Lagrangian cone structures via pseudo-product structures of type $G_{2}$. A local example of nonflat perturbations of the global model of flat Lagrangian cone structure which corresponds to $(2,3,5)$-distributions is given.

MSC classification

Primary: 58A30: Vector distributions (subbundles of the tangent bundles)
Secondary: 53A55: Differential invariants (local theory), geometric objects 53C17: Sub-Riemannian geometry 53A20: Projective differential geometry 70F25: Nonholonomic systems

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Type
Article
Information
Nagoya Mathematical Journal , Volume 243 , September 2021 , pp. 303 - 315
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Copyright
© 2020 Foundation Nagoya Mathematical Journal

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Footnotes

The first author was supported by JSPS KAKENHI No. 15H03615 and No. 15K13431.

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