Almost Geodesics and Special Affine Connection

In the present paper we continue to study almost geodesic curves and determine in R-n the form of curves C for which every image under an (n - 1)-dimensional algebraic torus is also an almost geodesic with respect to an affine connection del with constant coefficients. We also calculate explicitly the components of del. For the explicit calculation of the form of curves C in the n-dimensional real space Rn that are almost geodesics with respect to an affine connection del, we can suppose that with C all images of C under a real (n - 1)-dimensional algebraic torus are also almost geodesics. This implies that the determination of C becomes an algebraic problem. Here we use E. Beltrami's result that a differentiable curve is a local geodesic with respect to an affine connection del precisely if it is a solution of an abelian differential equation with coefficients that are functions of the components of del. Now we consider the special case for the connection del in which every curve is almost geodesic with respect to del.