quaternionic projective line$\,\mathbb{H}P^1$
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.
For $n \in \mathbb{N}_{\gt 0}$ and $r \in \mathbb{R}_{\gt 0}$, the Ricci tensor of the round $n$-sphere $S^n$ of radius $r$ satisfies
for all unit-length tangent vectors $v \in T S^n$, ${\vert v \vert} = 1$.
Accordingly, the scalar curvature of the round $n$-sphere of radius $r$ is the constant function with value
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)
Last revised on July 30, 2024 at 13:16:45. See the history of this page for a list of all contributions to it.