Hamiltonian point of view of non-Euclidean geometry and elliptic functions

✍ Scribed by V. Jurdjevic

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 179 KB
• Volume: 43
• Category: Article
• ISSN: 0167-6911
• DOI: 10.1016/s0167-6911(01)00093-7

✦ Synopsis

This paper o ers a new way of looking at the classical geometries and the theory of elliptic functions through Hamiltonian systems on Lie groups. In particular, the paper shows that: (i) the classical models of non-Euclidean geometries are canonically induced by bi-invariant sub-Riemannian metrics on Lie groups which act by left-actions on the underlying space; (ii) there is a class of canonical variational problems on Lie groups G whose projections on homogeneous spaces G=K generalize Euler's elasticae and include all curves of constant curvature and all -functions of Weierstrass; (iii) complex Lie groups unify non-Euclidean geometries and complex elasticae o er a distinctive look at the elliptic functions.