nLab 6-sphere




The n-sphere of dimension n=6n = 6.


Coset structure

The 6-sphere, as a smooth manifold is diffeomorphic to the coset space

S 6≃G 2/SU(3) S^6 \simeq G_2/ SU(3)

of G2 (automorphism group of the octonions) by SU(3) (Fukami-Ishihara 55).

For more see at G2/SU(3) is the 6-sphere.

The induced action of G2 on S 6S^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:

S nβˆ’1≃ diffSO(n)/SO(nβˆ’1)S^{n-1} \simeq_{diff} SO(n)/SO(n-1)this Prop.
S 2nβˆ’1≃ diffSU(n)/SU(nβˆ’1)S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)this Prop.
S 4nβˆ’1≃ diffSp(n)/Sp(nβˆ’1)S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)this Prop.
S 7≃ diffSpin(7)/G 2S^7 \simeq_{diff} Spin(7)/G_2Spin(7)/G2 is the 7-sphere
S 7≃ diffSpin(6)/SU(3)S^7 \simeq_{diff} Spin(6)/SU(3)since Spin(6) ≃\simeq SU(4)
S 7≃ diffSpin(5)/SU(2)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≃ diffG 2/SU(3)S^6 \simeq_{diff} G_2/SU(3)G2/SU(3) is the 6-sphere
S 15≃ diffSpin(9)/Spin(7)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)

Complex structure

A famous open problem is the question whether the 6-sphere admits an actual complex structure. For review see Bryant 14.


Last revised on December 1, 2019 at 19:19:37. See the history of this page for a list of all contributions to it.