Suppose \(\mathbb{S}^d_R\) is the \(d\)-sphere with radius \(R>0\) endowed with the round metric \(g\).
The Ricci curvature of \(\mathbb{S}^d_R\) is given by
\[ \Rc = \frac{d-1}{R^2}. \]In terms of any basis it is
\[ R_{ij}=\frac{d-1}{R^2}g_{ij}. \][1, Proposition 8.36]
References Link to heading
- J. Lee, Introduction to Riemannian Manifolds. Cham: Springer International Publishing, 2018. doi:10.1007/978-3-319-91755-9