Tangent bundle embeddings of manifolds in Euclidean space

Tangent Bundle Embeddings of Manifolds in Euclidean Space

For a given n-dimensional manifold M we study the problem of finding the smallest integer N(M) such that M admits a smooth embedding in the Euclidean space R without intersecting tangent spaces. We use the Poincaré-Hopf index theorem to prove that N(S) = 4, and construct explicit examples to show that N(S) ≤ 3n + 3, where S denotes the n-sphere. Finally, for any closed manifold M, we show that ...

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

Tangent Space Estimation for Smooth Embeddings of Riemannian Manifolds

Numerous dimensionality reduction problems in data analysis involve the recovery of lowdimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of the manifold at a point from locally available data samples. Local sampling conditions such as (i) the size of the neighborhood (sampling width) and (ii) the n...

Embeddings and Immersions of Manifolds in Euclidean Space

The problem of computing the number of embeddings or immersions of a manifold in Euclidean space is treated from a different point of view than is usually taken. Also, a theorem dealing with the existence of an embedding of Mm in ä ¡s given. Introduction. This paper studies the existence and classification of differential immersions and embeddings of a differentiable manifold into Euclidean spa...

Embeddings of Surfaces in Euclidean Three-space