Use the mobile referential method to calculate how curvatures (intrinsic, mean and extrinsic) of a great sphere in $\mathbb{S}^3$ (with canonical metric).
My teacher explained that a great sphere in $\mathbb{S}^3$ is analogous to a great circle in $\mathbb{S}^2$. You can take a definition of a vector subspace of dimension 3 within $\mathbb{R}^4$ with the sphere $\mathbb{S}^3$ within $\mathbb{R}^4$.
For example, if you use Cartesian coordinates $ x, y, z, w $ for $\mathbb{R}^4$ the equation of $\mathbb{S}^3$, $x^2 + y^2 + z^2 + w^2 = 1$, if you intersect $\mathbb{S}^3$ with the hyperplane defined by equation $w = 0$ you will get the unit sphere $x^2 + y^2 + z^2 = 1$ in $\mathbb{R}^3$ with coordinates $(x, y, z, 0)$.
I am not very well aware of the geometry involved and how to put these ideas on paper.