On the Osculating Spaces of Submanifolds in Euclidean Spaces

TL;DR

The paper addresses tight bounds on the dimensions of osculating spaces for $n$-dimensional manifolds embedded in Euclidean space $\mathbb{R}^m$. It introduces the higher-order normal curvature matrices $P^{(l)}$ and their eigenvectors to define successive normal vectors, proving a refined bound $k_r\le{n+r\choose r+1}$ and providing a geometric interpretation in terms of projections of iterated covariant derivatives. It then shows that the $r$-th osculating space is exactly the span of $Y_1,\dots,Y_n$ together with all relevant higher covariant derivatives up to order $r$, giving a constructive, finite description that tightens previous estimates. A sharpness argument via a specific embedding demonstrates that the bound is optimal. The results clarify the structure of osculating spaces and have implications for embedding geometry of submanifolds in Euclidean spaces.

Abstract

This paper is a continuation of the papers [2,3,4,5,6]. In this paper the osculating spaces of arbitrary order of a manifold embedded in Euclidean space are considered. A better estimation of their dimensions as well as the description of its basis are given.