Almost Geodesics and Special Affine Connection

Abstract

In the present paper we continue to study almost geodesic curves and determine in $${\mathbb {R}}^n$$ the form of curves $${\mathcal {C}}$$ for which every image under an $$(n-1)$$ -dimensional algebraic torus is also an almost geodesic with respect to an affine connection $$\nabla $$ with constant coefficients. We also calculate explicitly the components of $$\nabla $$ . For the explicit calculation of the form of curves $${\mathcal {C}}$$ in the n-dimensional real space $${\mathbb {R}}^n$$ that are almost geodesics with respect to an affine connection $$\nabla $$ , we can suppose that with $${{\mathcal {C}}}$$ all images of $${{\mathcal {C}}}$$ under a real $$(n-1)$$ -dimensional algebraic torus are also almost geodesics. This implies that the determination of $${{\mathcal {C}}}$$ 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 $$\nabla $$ precisely if it is a solution of an abelian differential equation with coefficients that are functions of the components of $$\nabla $$ . Now we consider the special case for the connection $$\nabla $$ in which every curve is almost geodesic with respect to $$\nabla $$ .