Equi-affine vector fields on manifold with equi-affine structure

Abstract

The aim is to extend the theory of affine plane curves on R 2 , to the manifolds of 2 and n dimensions. Affine arclength , affine curvatures, and equi-affine vector fields are defined on a manifold of dimension n with equi-affine structure. Classification of equi-affine vector fields on two dimensional equi-affine structure is obtained. Mathematics Subject Classification: 53A15

The equi-affine and Frenet curvatures of curves in pseudo-Riemannian 2-manifolds

“…They can be described as it follows e 2 , e 3 ), kp1 kp1 (5) κ 2 = εΩ(e 1 , ∇ e 1 e 3 , e 3 ) = εϕΩ(e 1 , ∇ α ′ e 3 , e 3 ). kp2 kp2 (6) Theorem 1. The equi-affine curvatures of a non-degenerate curve in a 3-dimensional equi-affine manifold can be expressed in the following way (8) Proof.…”

“…also J. Favard [7], K. Nomizu and T. Sasaki [17]. Next, such curves were investigated also in higher dimensional affine spaces by, among others, D. Davis [3] On the other hand, these ideas were extended to the non-degenerate curves in equi-affine manifolds in the papers by W. Barthel and A. Irmingard [1], M. Faghfouri and M. Toomanian [6], V. Hlavatý [10] and P. Stavre [21].…”

“…On the other hand, these ideas were extended to the non-degenerate curves in equi-affine manifolds in the papers by W. Barthel and A. Irmingard [1], M. Faghfouri and M. Toomanian [6], V. Hlavatý [10] and P. Stavre [21].…”