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

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 $S$.

Section 3 entirely rewritten, appendix made shorter, formula (8) in Theorem 2.1 for $\chi (S)$ in the older version had wrong sign now corrected, new results related to the Radon transform added