Some remarks on the moment curves in R4 and R3

Year 2025, Volume: 3 Issue: 1, 34 - 38, 24.06.2025

Mircea Crasmareanu

Abstract

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.

Keywords

moment curve , Hopf map , spherical curve

References

• Crasmareanu, M., 2016, Chin. Ann. Math., Ser. B, 37, no. 3, 419-426. google scholar
• Crasmareanu, M., 2021, Mat. Vesn., 73, no. 4, 256-267. google scholar
• Crasmareanu, M., 2024, Turkish J. Math., 48, no. 4, 749-756. google scholar
• Crasmareanu, M., 2024, Chapter in: Bang-Yen Chen et al (Eds.), "Geometry of Submanifolds and Applications", Infosys Science Foundation Series, Springer, 197-207. google scholar
• Edelman, A., Kostlan, E., 1995, Bull. Am. Math. Soc., 32, no. 1, 1-37. google scholar
• Takahashi, M., 2016, In: Nabarro, Ana Claudia et al (Eds.), "Real and complex singularities", AMS Contemporary Mathematics 675, 337-355. google scholar