6-sphere in nLab

Contents

Contents

Idea

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

Properties

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

Complex structure

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

• fuzzy 6-sphere

• 2-sphere

• 3-sphere

• 4-sphere

• 7-sphere

• n-sphere

References

• T. Fukami, S. Ishihara, Almost Hermitian structure on S 6S^6 , Tohoku Math J. 7 (1955), 151–156.

• Ilka Agricola, Aleksandra Borówka, Thomas Friedrich, S 6S^6 and the geometry of nearly Kähler 6-manifolds (arXiv:1707.08591)

• Robert Bryant, S.-S. Chern’s 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)