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 and Travis WILLSE

Basic information

Original name

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

Authors

GOVER, A. Rod, Katharina NEUSSER (40 Austria, guarantor, belonging to the institution) and Travis WILLSE

Edition

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

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Germany

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 1.000 in 2022

Organization unit

Faculty of Science

DOI

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

UT WoS

001091151700001

Keywords in English

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

Abstract

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.

Links

GA19-06357S, research and development project

Name: Geometrické struktury, diferenciální operátory a symetrie (Acronym: GSDOS)

Investor: Czech Science Foundation

Citovat

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

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