Should I consider a 1-dimensional submanifold and revolve it to obtain a $n$-dimensional manifold or it should be of specific dimension?
For example I want to consider $\Bbb S^4$ as manifold of revolution. (More specifically I want to work out with 4-ellipsoid). What I should do? I should start with a semi-circle or a half-sphere or a half 3-sphere? in 3 dimensions, it is too classic that $x(t, \theta) = (a(t)\cos\theta, a(t)\sin\theta, b(t))$ for a curve $(a(t),b(t))$ embedded in $zy$-plane.
In next step I want to calculate its Riemann curvature tensor and related quantities.