Suppose \(\mathbb{S}^d_R\) is the \(d\)-sphere with radius \(R>0\) endowed with the round metric \(g\).

The Riemann curvature endomorphism of \(\mathbb{S}^d_R\) is given by

\[ R(v,w)x = \tfrac{1}{R^2} (\langle w, x\rangle v - \langle v, x\rangle w). \]

In terms of any basis it is

\[ R_{ijkl} = \tfrac{1}{R^2} (g_{il}g_{jk} - g_{ik}g_{jl}) \]

[1, Proposition 8.36].

Remark
  • The curvature endomorphism is covariantly constant, since \(g\) is covariantly constant.

References Link to heading

  1. J. Lee, Introduction to Riemannian Manifolds. Cham: Springer International Publishing, 2018. doi:10.1007/978-3-319-91755-9