Higher-order normals on manifolds

On an $n$-dimensional smooth manifold, we consider higher-order normals of two types, i.e., the spaces that complement the tangent space of orders $1$ or ${r-1}$ to the tangent space of order $r$. We prove that the derivatives of some basic vectors in the direction of the given first-order (second-order) basis vectors are equal to the values on these vectors of the first-order (second-order) differentials of the first vectors. Using the differentials of basic tangent vectors of the first and second orders, we construct mappings from the set of first-order tangent vectors to the set of second- and third-orders normal vectors. Also, we introduce mappings that generate horizontal second- and third-order vectors for the canonical first- and second- order affine connections, respectively.