Gravity and Strings

Appendix C
n-Spheres
An n-dimensional unit-radius 1 sphere S n is the hypersurface of R n+1 defined by
(x 1 ) 2 + ···+(x n+1 ) 2 = 1. It is usually parametrized in terms of spherical coordinates
{r,ϕ,θ1,...,θn−1},
where
x 1 = ρn−1 sin ϕ,
x 2 = ρn−1 cos ϕ,
x 3 = ρn−2 cos θ1, (C.1)
.
.
x k = ρn−k+1 cos θk−2, 3 ≤ k ≤ n + 1,
ρl = [(x 1 ) 2 + ···+(x n+1−l ) 2 ] 1 2 = r
ρ0 = r = [(x 1 ) 2 + ···+(x n+1 ) 2 ] 1 2 ,
l
sin θn−m,
m=1
(C.2)
and ϕ ∈ [2, 2π], θi ∈ [0,π], setting r = 1. The metric induced on Sn in spherical coordinates
is denoted by d2 (n) and is implicitly defined in
d x 2
(n+1) = dρ2 0 + ρ2 0dθ 2 n−1 + ···+ρ2 n−2dθ 2 1 + ρ2 n−1dϕ2 ≡ dr 2 + r 2 d 2 (n) . (C.3)
In practice, it is convenient to use the recursive formula
d 2 (n) = dθ 2 n−1 + sin2 θn−1 d 2 (n−1) , d2 (1) = dϕ2 . (C.4)
The spheres equipped with this metric, which is clearly SO(n + 1)-invariant, are called
round spheres (see Appendix A.4.2). Other metrics with less symmetry on the same S n
manifolds are possible, but sometimes a different notation is used to denote the corresponding
Riemannian spaces.
1 Foratopological space, the radius is irrelevant, but it becomes relevant when we consider the metric induced
from the Euclidean metric of Rn+1 .
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