quaternionic projective line$\,\mathbb{H}P^1$
The n-sphere of dimension $n = 6$.
The 6-sphere, as a smooth manifold is diffeomorphic to the coset space
of Gβ (automorphism group of the octonions) by SU(3) (Fukami-Ishihara 55).
For more see at Gβ/SU(3) is the 6-sphere.
The induced action of Gβ on $S^6$ induces an almost Hermitian structure which makes it a nearly Kaehler manifold?.
Review in is in Agrikola-Borowka-Friedrich 17
coset space-structures on n-spheres:
standard: | |
---|---|
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$ | this Prop. |
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$ | this Prop. |
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$ | this Prop. |
exceptional: | |
$S^7 \simeq_{diff} Spin(7)/G_2$ | Spin(7)/Gβ is the 7-sphere |
$S^7 \simeq_{diff} Spin(6)/SU(3)$ | since Spin(6) $\simeq$ SU(4) |
$S^7 \simeq_{diff} Spin(5)/SU(2)$ | since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere |
$S^6 \simeq_{diff} G_2/SU(3)$ | Gβ/SU(3) is the 6-sphere |
$S^15 \simeq_{diff} Spin(9)/Spin(7)$ | Spin(9)/Spin(7) is the 15-sphere |
see also Spin(8)-subgroups and reductions
homotopy fibers of homotopy pullbacks of classifying spaces:
(from FSS 19, 3.4)
A famous open problem is the question whether the 6-sphere admits an actual complex structure. For review see Bryant 14.
T. Fukami, S. Ishihara, Almost Hermitian structure on $S^6$ , Tohoku Math J. 7 (1955), 151β156.
Ilka Agricola, Aleksandra BorΓ³wka, Thomas Friedrich, $S^6$ and the geometry of nearly KΓ€hler 6-manifolds (arXiv:1707.08591)
Robert Bryant, S.-S. Chernβs study of almost-complex structures on the six-sphere (arXiv:1405.3405)
Robert Bryant, Remarks on the geometry of almost complex 6-manifolds (arXiv:math/0508428)
Last revised on July 18, 2024 at 10:38:19. See the history of this page for a list of all contributions to it.