We consider a standard Determinantal Point Process in the complex affine space of any given dimension. Then, we describe how such a process can be transported to another manifold via diffeomorphism. The most natural such manifold is the complex projective space.

This gives us a collection of quite well-distributed complex projective points, which we lift to the unit sphere of any odd dimension using the Hopf fibration. A number of equispaced points is chosen in each fiber (the fibers are just circles), and if the number of points in the fiber is carefully chosen it turns out that the obtained set of spherical points have a very small energy in several senses. This is a step towards understanding constructive collections of low-energy points in odd-dimensional spheres.