On a manifold diffeomorphic to $\mathbb{S}^n$, the most canonical metric is the round one. It is also known that there are a large number of Einstein metrics on spheres which are not round. Are there explicit canonical metrics on $\mathbb{S}^n$, however? For example, explicit metrics of constant scalar curvature? Explicit Einstein metrics?
By explicit, I mean that I have an explicit local coordinate expression for the metric.
I am interested only in the case $n \geq 3$.
