On the osculating spaces of submanifolds in Euclidean spaces (Q6486667)

The author considers osculating spaces of arbitrary order of an \(n\)-dimensional manifold \(M\) embedded in the Euclidean space \(\mathbb R^m\) and gives the best possible estimation of their dimension. In particular, he proves that the \(r\)-dimensional osculating space coincides with the special space generated by the orthonormal basis \(\{Y_1,Y_2,\dots,Y_n\}\) of the tangent space and the vectors \(\{\nabla_{Y_i}Y_j\}\) (\(i\leq j\)), \(\{\nabla_{Y_i}\nabla_{Y_j}Y_p\}\) (\(i\leq j\leq p\)),\dots, \(\{\nabla_{Y_{i_1}}\nabla_{Y_{i_2}}\cdots\nabla_{Y_{i_{r-1}}}Y_{i_r}\}\) (\(i_1\leq i_2\leq\dots\leq i_{r}\)).