# Newton's 1st law, applied in a manifold

A seemingly interesting question.

Newton's 1st law states that objects continue in straight lines, unless acted upon by external forces. Now consider a frictionless manifold. Since it is locally euclidian, the object will travel in a straight line. In other words, if we build a frictionless track surrounding the earth, neglect all other forces and air, will the object be able to circle the earth unless acted upon by external forces?

Yes, "straight" lines on a manifold are geodesics, which obey the geodesic equation $$\frac{\text{d}^2x^\rho}{\text{d}\tau^2} = -\Gamma^\rho_{\mu\nu}\frac{\text{d}x^\mu}{\text{d}\tau}\frac{\text{d}x^\nu}{\text{d}\tau}$$