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 $n$-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*.

**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)/G2 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 |

$S^6 \simeq_{diff} G_2/SU(3)$ | G2/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)

Last revised on April 25, 2019 at 10:12:57. See the history of this page for a list of all contributions to it.