Hausdorff Dimension and Lebesgue Measure of Codiagonal of Embedded Vector Bundles over Submanifolds in Euclidean Space

TL;DR

This paper investigates the measure-theoretic and geometric properties of the codiagonal images of embedded vector bundles over submanifolds in Euclidean space, revealing conditions for openness, positive measure, and maximal Hausdorff dimension.

Contribution

It establishes new criteria for the interior and measure of codiagonals of vector bundles, including tangent and normal bundles, under weak smoothness assumptions.

Findings

Normal bundles' codiagonals contain open sets under weak smoothness.

Codiagonals of tangent bundles of hypersurfaces have non-empty interior unless contained in a hyperplane.

Union of hyperplanes covering a hypersurface has maximal Hausdorff dimension.

Abstract

In this paper we study measure theoretical size of the image of naturally embedded vector bundles in $\mathbb{R}^{n} \times \mathbb{R}^{n}$ under the codiagonal morphism, i.e. $\Delta_{*}$ in the category of finite dimensional $\mathbb{R}$-vector spaces. Under very weak smoothness condition we show that codiagonal of normal bundles always contain an open subset of the ambient space, and we give corresponding criterions for the tangent bundles. For any differentiable hypersurface we show that the codiagonal of its tangent bundle has non-empty interior, unless the hypersurface is contained in a hyperplane. Assuming further smoothness (e.g. twice differentiable) we show that union of any family of hyperplanes that covers the hypersurface has maximal possible Hausdorff dimension. We also define and study a notion of degeneracy of embedded $C^{1}$ vector bundles over a $C^{1}$ submanifold and show as a corollary that if the base manifold has at least one non-inflection point then codiagonal of any $C^{1}$ line bundle over it has positive Lebesgue measure. Finally we show that codiagonal of any line bundle over an $n$-dimensional ellipsoid or a convex curve has non-empty interior, and the same assertion also holds for any non-tangent line bundle over a hyperplane.