Multidimensional generalizations of Jacobi's envelope theorem

In the paper, the extension of a field of geodesics $\wp$ on a manifold $N$ isometrically embedded in a Riemannian manifold $M$ is considered. The symplectic involute of the manifold N along the field $\wp$ is defined and a theorem is proved which gives a multidimensional analog of Jacobi's envelope theorem.