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A HYDRODYNAMICAL HOMOTOPY CO-MOMENTUM MAP AND A MULTISYMPLECTIC INTERPRETATION OF HIGHER-ORDER LINKING NUMBERS

Part of: Operations and obstructions Symplectic geometry, contact geometry

Published online by Cambridge University Press:  11 February 2021

ANTONIO MICHELE MITI Open the ORCID record for ANTONIO MICHELE MITI [Opens in a new window]  and
MAURO SPERA Open the ORCID record for MAURO SPERA [Opens in a new window]
ANTONIO MICHELE MITI*
Affiliation:
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121Brescia, Italy and Department of Mathematics, KU Leuven, Celestijnenlaan 200B Box 2400, 3001Leuven, Belgium
MAURO SPERA
Affiliation:
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121Brescia, Italy e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

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In this paper a homotopy co-momentum map (à la Callies, Frégier, Rogers and Zambon) transgressing to the standard hydrodynamical co-momentum map of Arnol’d, Marsden, Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coadjoint orbits associated to the Euler evolution for perfect fluids, and in particular of Brylinski’s manifold of smooth oriented knots, is discussed. As an application of the above homotopy co-momentum map, a reinterpretation of the (Massey) higher-order linking numbers in terms of conserved quantities within the multisymplectic framework is provided and knot-theoretic analogues of first integrals in involution are determined.

Keywords

higher-order linking numbershomotopy co-momentum mapshydrodynamicssymplectic and multisymplectic geometry

MSC classification

Primary: 53D20: Momentum maps; symplectic reduction
Secondary: 53D99: None of the above, but in this section 55S30: Massey products
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 3 , June 2022 , pp. 335 - 354
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Footnotes

Communicated by Robert Yuncken

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