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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F24%3A00139365" target="_blank" >RIV/00216224:14310/24:00139365 - isvavai.cz</a>

Výsledek na webu

<a href="https://link.springer.com/article/10.1007/s10231-023-01385-0" target="_blank" >https://link.springer.com/article/10.1007/s10231-023-01385-0</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s10231-023-01385-0" target="_blank" >10.1007/s10231-023-01385-0</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA19-06357S" target="_blank" >GA19-06357S: Geometrické struktury, diferenciální operátory a symetrie</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Annali di Matematica Pura ed Applicata

ISSN

0373-3114

e-ISSN

1618-1891

Svazek periodika

203

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

28

Strana od-do

875-902

Kód UT WoS článku

001091151700001

EID výsledku v databázi Scopus

2-s2.0-85174815078