A note on constant geodesic curvature curves on surfaces

Poleni Curves on Surfaces of Constant Curvature

In the euclidean plane, a regular curve can be defined through its intrinsic equation which relates its curvature k to the arc length s. Elastic plane curves were determined this way. If k(s) = 2α cosh(αs) , the curve is known by the name “la courbe des forçats”, introduced in 1729 by Giovanni Poleni in relation with the tractrix [9]. The above equation is yet meaningful on a surface if one int...

Geodesic curves on Shimura surfaces

A Shimura surface is the quotient of either the product H×H of two hyperbolic planes or the unit ball HC in C by an irreducible arithmetic lattice. Examples include the normal quasiprojective varieties associated with the Hilbert and Picard modular groups, along with the solutions to many moduli problems for principally polarized abelian varieties. Special amongst the immersed projective algebr...

Curves on surfaces: Geodesics 1 Review of the geodesic curvature of the curves in the surface

1 Review of the geodesic curvature of the curves in the surface We first review the geodesic curvature of the curves in the surface introduced in section 4.1. Let M be a surface and let α be a curve in the surface M, parametrized by its arc-length (.i.e. α ′ = 1). By the " trick of the unit-vector " , α ′′ is perpendicular to α ′ , so (4.2.1) α ′′ = κ n n + κ g n × α ′ .

Topology Proceedings GEODESIC CURVES ON SHIMURA SURFACES

A Shimura surface is the quotient of either the product H×H of two hyperbolic planes or the unit ball HC in C by an irreducible arithmetic lattice. Examples include the normal quasiprojective varieties associated with the Hilbert and Picard modular groups, along with the solutions to many moduli problems for principally polarized abelian varieties. Special amongst the immersed projective algebr...

Constant scalar curvature metrics on toric surfaces