2019/02/28 by Shimpei Kobayashi, Takeshi Sasaki, Kobayashi, Shimpei +1
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1902.10926
openalex publication_date 2019/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups \rm GA(2)=\rm GL(2,\bf R)\ltimes \bf R2 and \rm GA(3)=\rm GL(3,\bf R)\ltimes \bf R3, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves.