• DOI:10.1007/s00229-019-01114-z
• Corpus ID: 119159201

A Note on Generic Transversality of Euclidean Submanifolds

@article{Li2018ANO,
  title={A Note on Generic Transversality of Euclidean Submanifolds},
  author={Siran Li},
  journal={manuscripta mathematica},
  year={2018},
  volume={162},
  pages={213-219}
}

• Siran Li
• Published 3 November 2018
• Mathematics
• manuscripta mathematica

In this short note, we establish a quantitative description of the genericity of transversality of $C^1$-submanifolds in $\mathbb{R}^n$: Let $\Sigma \subset \mathbb{R}^n$ be a $d$-dimensional $C^1$-embedded submanifold where $n \geq d+1$. Denote by \begin{equation} \mathscr{A}(\Sigma) := \bigg\{ a \in \mathbb{R}^n: {\rm volume}\,\Big\{ p\in\Sigma : \partial\mathbb{B}(a, |a-p|) \text{ is not transversal to $\Sigma$ at $p$} \Big\} > 0 \bigg\}. \end{equation} Then $\mathscr{A}(\Sigma)$ is contained in a countable union of $(n-d-1)$-dimensional affine planes.