The Euler characteristic of a surface from its Fourier analysis in one direction

TL;DR

This paper demonstrates that the genus and Euler characteristic of a closed surface in three-dimensional space can be recovered from the Fourier transform and wave equation solutions restricted to lines, linking geometric invariants to Fourier analysis.

Contribution

It introduces a novel method to determine topological invariants of surfaces using Fourier analysis and wave equations, connecting geometric properties with harmonic analysis techniques.

Findings

Genus of a surface can be recovered from Fourier transform restrictions.

Euler characteristic can be obtained from wave equation solutions.

Method applies to generic lines in space and Minkowski space.

Abstract

In this paper, we prove that we can recover the genus of a closed compact surface $S$ in $\mathbb{R}^3$ from the restriction to a generic line of the Fourier transform of the canonical measure carried by $S$. We also show that the restriction on some line in Minkowski space of the solution of a linear wave equation whose Cauchy data comes from the canonical measure carried by $S$, allows to recover the Euler characteristic of $S$.