Removing the a priori restrictions in the theorem on a complete system of invariants of a curve in \(\mathbb E^{n}_{l}\) (Q1363469)

For a pair of integers \( n\geq 2 \), \( l<n \), a segment \( I \subset \mathbb R^1 \), and \( n-1 \) functions \( \varphi_1 (t) , \dots ,\varphi_{n-1}(t) \) such that \( \varphi_i (t) > 0 \) for \( i< n-1 \) and \( \varphi_{n-1} (t) \geq 0 \), \( t \in I \), the author establishes necessary and sufficient conditions for existence of a naturally parametrized curve \( \gamma \) in a pseudo-Euclidean space \( \mathbb E^{n}_{l} \) which has curvatures of the orders \( 1, \dots, n-1 \) equal to \( \varphi_1 (t) , \dots, \varphi_{n-1}(t) \), respectively. It is also proven that any two such curves \( \gamma \) and \( \gamma^* \) are congruent if and only if the set \(\{ t \in I \mid \varphi_{n-1} (t) >0 \} \) is connected. Here \( n \) is the dimension and \( l \) is the index of \( \mathbb E^{n}_{l} \). A similar result on existence of a curve in the Euclidean space \( \mathbb E^n \) with prescribed integral curvatures is also obtained.