  Let $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$
denote the set of $C^1$ regular curves in the $2$-sphere $\mathbb{S}^2$ that
start and end at given points with the corresponding Frenet frames
$\boldsymbol{P}$ and $\boldsymbol{Q}$, whose tangent vectors are Lipschitz
continuous, and their a.e. existing geodesic curvatures have essentially bounds
in $(\kappa_1, \kappa_2)$, $-\infty<\kappa_1<\kappa_2<\infty$. In this article,
firstly we study the geometric property of the curves in
$\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$. We
introduce the concepts of the lower and upper curvatures at any point of a
$C^1$ regular curve and prove that a $C^1$ regular curve is in
$\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ if and only
if the infimum of its lower curvature and the supremum of its upper curvature
are constrained in $(\kappa_1,\kappa_2)$. Secondly we prove that the $C^0$ and
$C^1$ topologies on $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P},
\boldsymbol{Q})$ are the same. Further, we show that a curve in
$\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ can be
determined by the solutions of differential equation $\Phi'(t) =
\Phi(t)\Lambda(t)$ with $\Phi(t)\in \textrm{SO}_3(\mathbb{R})$ with special
constraints to $\Lambda(t)\in\mathfrak{so}_3(\mathbb{R})$ and give a complete
metric on $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$
such that it becomes a (trivial) Banach manifold.
