This is a definition I encountered in a paper. I hope someone will be able to help me understand it. The authors assume a Frenet curve $\alpha(s)$ on a 3-D Riemannian Manifold as any non-geodesic unit speed-curve that satisfies the Frenet Equations using the Levi-Civita Connection $\nabla_s$, $s$ being the arc length parameter of the curve $\alpha$. Then they give the following definition(part of which is given): Let $α(s)$ be a Frenet curve in a 3-dimensional Riemannian manifold M and $\lbrace T_α,N_α,B_α\rbrace$ the Frenet frame of α. Consider a surface $X_{N_\alpha}$ defined by $$X_{N_\alpha}(s, t) = exp_{α(s)}(tN_α(s)).$$
I am interested in understanding the nature of this surface.
I presume that as defined in the same paper: $exp_p(v)=\gamma_v(1)$ for any $v\in T_p(M)$,where $\gamma_v$ is a constant speed geodesic on M defined on $[0,\infty)$ such that $\gamma_v(0)=p,\gamma_v'(0)=v$ , the above surface could be written as $X_{N_\alpha}(s,t)=\gamma_{N_\alpha(s)}(t)$, but that is not helping me much.
I would also like to know why the definition of the exponential map here requires the geodesic to be of constant speed.