round sphere in nLab

Proposition

For n∈ℕ >0n \in \mathbb{N}_{\gt 0} and r∈ℝ >0r \in \mathbb{R}_{\gt 0}, the Ricci tensor of the round n n -sphere S nS^n of radius rr satisfies

Ric(v,v)=n−1r 2 Ric(v,v) \;=\; \frac{n-1}{r^2}

for all unit-length tangent vectors v∈TS nv \in T S^n, |v|=1{\vert v \vert} = 1.

Accordingly, the scalar curvature of the round n n -sphere of radius rr is the constant function with value

R=n(n−1)r 2. \mathrm{R} \;=\; \frac{n(n-1)}{r^2} \,.