Author:
Boydon Kai Brynne M.,Piccinni Paolo
Abstract
AbstractA comparison among different constructions in $$\mathbb {H}^2 \cong {\mathbb {R}}^8$$
H
2
≅
R
8
of the quaternionic 4-form $$\Phi _{\text {Sp}(2)\text {Sp}(1)}$$
Φ
Sp
(
2
)
Sp
(
1
)
and of the Cayley calibration $$\Phi _{\text {Spin}(7)}$$
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 $$\text {Spin}(7)$$
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 $$\mathbb {R}^{16}$$
R
16
for the canonical 4-forms $$\Phi _{\text {Spin}(8)}$$
Φ
Spin
(
8
)
and $$\Phi _{\text {Spin}(7)\text {U}(1)}$$
Φ
Spin
(
7
)
U
(
1
)
, associated with Clifford systems related with the subgroups $$\text {Spin}(8)$$
Spin
(
8
)
and $$\text {Spin}(7)\text {U}(1)$$
Spin
(
7
)
U
(
1
)
of $$\text {SO}(16)$$
SO
(
16
)
. We characterize the calibrated 4-planes of the 4-forms $$\Phi _{\text {Spin}(8)}$$
Φ
Spin
(
8
)
and $$\Phi _{\text {Spin}(7)\text {U}(1)}$$
Φ
Spin
(
7
)
U
(
1
)
, extending in two different ways the notion of Cayley 4-plane to dimension 16.
Funder
Università degli Studi di Roma La Sapienza
Publisher
Springer Science and Business Media LLC
Cited by
1 articles.
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1. Clifford systems, Clifford structures, and their canonical differential forms;Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg;2020-12-08