Some remarks on the moment curves in R4 and R3

The moment curves and their normalizations are key tools in obtaining the famous Kac formula from the theory of random polynomials. We study here the normalized moment curves Γ𝑛 ∈ 𝑆𝑛 in the low dimensions 𝑛 where 𝑆𝑛 is the Euclidean 𝑛 dimensional unit sphere; more precisely we consider 𝑛 = 3 and 𝑛 = 2. First, we compute the image of the normalized moment curve Γ3 under the well-known Hopf fibre map and show that this remarkable map reduces the length of Γ3. Second, we analyze the curve Γ2 using the theory of spherical Legendre curves. An image of Γ2 is included.