Recently, I stumbled upon the definition of regular homotopy classes and Smale's Theorem as in Smale, S. (1958). Regular curves on Riemannian manifolds. Transactions of the American Mathematical Society, 87(2), 492-512. In particular, this result gives a one-to-one correspondance between the regular homotopy classes $\pi_R(M)$ of a Riemaniann Manifold $M$ and the first homotopy group of the tangent bundle of $M$, $\pi_1(T(M))$.
I was wondering, is there a (natural?) group structure on $\pi_R(M)$? I would really appreciate a reference as I'm more an analysis guy and I'm not very proficient in Geometry.
This question arises from trying to understand if the total curvature of a given smooth, simple and closed curve $\gamma$ on the 2-dimensional (flat) torus can only be $0$ or $\pm2\pi$. One one hand I know that the total curvature is invariant under regular homotopy. On the other hand any curve $\gamma$ as above is homotopic (or more, isotopic) to a line of rational slope on the cube $Q=[0,1)^2$. I was hoping of bridging these two facts to conclude, as the homotopy classes can be identified with union of vertical and horizontal lines in $Q$.