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Let P κ2 κ1 (P , Q) denote the set of C 1 regular curves in the 2-sphere S 2 that start and end at given points with the corresponding Frenet frames P and Q, whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in (κ 1 , κ 2 ), −∞ < κ 1 < κ 2 < ∞. In this article, firstly we study the geometric property of the curves in P κ2 κ1 (P , 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 P κ2 κ1 (P , Q) if and only if the infimum of its lower curvature and the supremum of its upper curvature are constrained in (κ 1 , κ 2 ). Secondly we prove that the C 0 and C 1 topologies on P κ2 κ1 (P , Q) are the same. Further, we show that a curve in P κ2 κ1 (P , Q) can be determined by the solutions of differential equation Φ ′ (t) = Φ(t)Λ(t) with Φ(t) ∈ SO 3 (R) with special constraints to Λ(t) ∈ so 3 (R) and give a complete metric on P κ2 κ1 (P , Q) such that it becomes a (trivial) Banach manifold.Contents 20 References 24

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