Compactifications of indefinite 3-Sasaki structures and their
quaternionic Kähler quotients

J 2024

Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients

GOVER, A. Rod, Katharina NEUSSER a Travis WILLSE

Základní údaje

Originální název

Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients

Autoři

GOVER, A. Rod, Katharina NEUSSER (40 Rakousko, garant, domácí) a Travis WILLSE

Vydání

Annali di Matematica Pura ed Applicata, Springer, 2024, 0373-3114

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Německo

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 1.000 v roce 2022

Organizační jednotka

Přírodovědecká fakulta

DOI

http://dx.doi.org/10.1007/s10231-023-01385-0

UT WoS

001091151700001

Klíčová slova anglicky

Projective differential geometry; Einstein manifolds; Sasaki manifolds; Hyper-Kähler and quaternionic Kähler geometry; Holonomy; Geometric compactifications

Štítky

rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 12. 3. 2024 14:34, Mgr. Marie Šípková, DiS.

Anotace

V originále

We show that 3-Sasaki structures admit a natural description in terms of projective differential geometry. First we establish that a 3-Sasaki structure may be understood as a projective structure whose tractor connection admits a holonomy reduction, satisfying a particular non-vanishing condition, to the (possibly indefinite) unitary quaternionic group Sp(p, q). Moreover, we show that, if a holonomy reduction to Sp(p, q) of the tractor connection of a projective structure does not satisfy this condition, then it decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) 3-Sasaki structures and a closed separating hypersurface at infinity with respect to the 3-Sasaki metrics. It is shown that the latter hypersurface inherits a Biquard–Fefferman conformal structure, which thus (locally) fibers over a quaternionic contact structure, and which in turn compactifies the natural quaternionic Kähler quotients of the 3-Sasaki structures on the open manifolds. As an application, we describe the projective compactification of (suitably) complete, non-compact (indefinite) 3-Sasaki manifolds and recover Biquard’s notion of asymptotically hyperbolic quaternionic Kähler metrics.

Návaznosti

GA19-06357S, projekt VaV

Název: Geometrické struktury, diferenciální operátory a symetrie (Akronym: GSDOS)

Investor: Grantová agentura ČR, Geometrické struktury, diferenciální operátory a symetrie

Citovat

GOVER, A. Rod, Katharina NEUSSER a Travis WILLSE. Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients. Annali di Matematica Pura ed Applicata. Springer, 2024, roč. 203, April 2024, s. 875-902. ISSN 0373-3114. Dostupné z: https://dx.doi.org/10.1007/s10231-023-01385-0.

@article{2374058,
   author = {Gover, A. Rod and Neusser, Katharina and Willse, Travis},
   article_number = {April 2024},
   doi = {http://dx.doi.org/10.1007/s10231-023-01385-0},
   keywords = {Projective differential geometry; Einstein manifolds; Sasaki manifolds; Hyper-Kähler and quaternionic Kähler geometry; Holonomy; Geometric compactifications},
   language = {eng},
   issn = {0373-3114},
   journal = {Annali di Matematica Pura ed Applicata},
   title = {Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients},
   url = {https://link.springer.com/article/10.1007/s10231-023-01385-0},
   volume = {203},
   year = {2024}
}

TY  - JOUR
ID  - 2374058
AU  - Gover, A. Rod - Neusser, Katharina - Willse, Travis
PY  - 2024
TI  - Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients
JF  - Annali di Matematica Pura ed Applicata
VL  - 203
IS  - April 2024
SP  - 875-902
EP  - 875-902
PB  - Springer
SN  - 03733114
KW  - Projective differential geometry
KW  - Einstein manifolds
KW  - Sasaki manifolds
KW  - Hyper-Kähler and quaternionic Kähler geometry
KW  - Holonomy
KW  - Geometric compactifications
UR  - https://link.springer.com/article/10.1007/s10231-023-01385-0
N2  - We show that 3-Sasaki structures admit a natural description in terms of projective differential geometry. First we establish that a 3-Sasaki structure may be understood as a projective structure whose tractor connection admits a holonomy reduction, satisfying a particular non-vanishing condition, to the (possibly indefinite) unitary quaternionic group Sp(p, q). Moreover, we show that, if a holonomy reduction to Sp(p, q) of the tractor connection of a projective structure does not satisfy this condition, then it decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) 3-Sasaki structures and a closed separating hypersurface at infinity with respect to the 3-Sasaki metrics. It is shown that the latter hypersurface inherits a Biquard–Fefferman conformal structure, which thus (locally) fibers over a quaternionic contact structure, and which in turn compactifies the natural quaternionic Kähler quotients of the 3-Sasaki structures on the open manifolds. As an application, we describe the projective compactification of (suitably) complete, non-compact (indefinite) 3-Sasaki manifolds and recover Biquard’s notion of asymptotically hyperbolic quaternionic Kähler metrics.
ER  -

GOVER, A. Rod, Katharina NEUSSER a Travis WILLSE. Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients. \textit{Annali di Matematica Pura ed Applicata}. Springer, 2024, roč.~203, April 2024, s.~875-902. ISSN~0373-3114. Dostupné z: https://dx.doi.org/10.1007/s10231-023-01385-0.

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