Clifford systems, Clifford structures, and their canonical differential forms
A comparison among different constructions in H 2 ≅ R 8 of the quaternionic 4-form Φ Sp ( 2 ) Sp ( 1 ) and of the Cayley calibration Φ Spin ( 7 ) shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in Spin ( 7 ) geometry. This compa...
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| Veröffentlicht in: | Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2021-04, Vol.91 (1), p.101-115 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | A comparison among different constructions in
H
2
≅
R
8
of the quaternionic 4-form
Φ
Sp
(
2
)
Sp
(
1
)
and of the Cayley calibration
Φ
Spin
(
7
)
shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in
Spin
(
7
)
geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension 16, similar constructions allow to write explicit formulas in
R
16
for the canonical 4-forms
Φ
Spin
(
8
)
and
Φ
Spin
(
7
)
U
(
1
)
, associated with Clifford systems related with the subgroups
Spin
(
8
)
and
Spin
(
7
)
U
(
1
)
of
SO
(
16
)
. We characterize the calibrated 4-planes of the 4-forms
Φ
Spin
(
8
)
and
Φ
Spin
(
7
)
U
(
1
)
, extending in two different ways the notion of Cayley 4-plane to dimension 16. |
|---|---|
| ISSN: | 0025-5858 1865-8784 |
| DOI: | 10.1007/s12188-020-00229-5 |