A generalization of Clairaut's formula and its applications

The main purpose of this article is to study conditions for a curve on a submanifold $M\subset {\mathbb{R}}^n$, constructed in a particular way involving the Euclidean distance to $M$, to be a geodesic. We also present the naturally arising generalization of Clairaut’s formula needed for the generalization of the main result to higher dimensions.