I have a regular curve $\alpha(s)$ on a 3-D Riemannian Manifold, with Frenet frame $\{T,N,B \}$. I define a surface now as
$$X_N(s,t) = exp_{\alpha(s)}(tN(\alpha(s)))$$
$exp_p$ being the exponential map at $p \in M$. I think the surface is a ruled surface atleast in $\mathbb R^3$. Can anyone tell me whether it will be something similar in a general case, in particular for $\mathbb S^3$ and the 3-D Hyperbolic plane equipped with the Poincare metric??
Also in case of $\mathbb R^3$, I have $X_N(s,t) = \alpha(s) + t N(\alpha(s))$, I dont know whether the sum makes sense as $N(s)$ need not be in $\mathbb R^3$?? So should I be seeing the whole thing in $\mathbb R^4$ ??
It is ruled, in the sense that it is fibered by geodesics. You need to assume that the tangent to $\alpha$ is nowhere parallel, $\nabla_T T\neq 0$, so that $N=\nabla_T T/\|\nabla_T T\|$ makes sense. Your formula $X_N(s,t) = exp_{\alpha(s)}(tN(\alpha(s)))$ then defines an immersed surface, at least in a neighborhood of $t=0$, fibered by the geodesics $s=const.$.