nLab round sphere




In Riemannian geometry, the topological n-sphere regarded as a Riemannian manifold in the standard way (i.e. as the submanifold of elements at constant distance from a given point in Euclidean space) is also called the round nn-sphere, in order to distinguish it from other, non-isometric Riemannian manifold structures that also exists on some n-sphere. These alternatives are then also called squashed spheres.

Examples of squashed nn-spheres

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)/Gβ‚‚ 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)Gβ‚‚/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)

Last revised on December 1, 2019 at 19:10:14. See the history of this page for a list of all contributions to it.