Combing a hedgehog over a field
Alexey Ananyevskiy, Marc Levine
TL;DR
This work establishes an algebro-geometric criterion for the existence of non-vanishing tangent fields on smooth affine quadrics $Q^o$ over perfect fields $k$ with $\operatorname{char} k\neq 2$, by translating the problem into Chow-Witt theory and quadratic form arithmetic. The authors show that for odd $n$ the tangent bundle $T_{Q^o}$ always admits a non-vanishing section, while for even $n$ the obstruction is controlled by $-1$ in the norm/group data $[D(q)]$ and $[D(q)^2]$, via a computable Euler-class criterion and Scharlau transfers. Central to the approach are the quadratic Gauss–Bonnet theorem in $GW(k)$ and the Euler class $e(T_{Q^o})$ living in Chow-Witt groups; these yield explicit criteria and connect to Pfister forms, transfer ideals, and norm principles. The paper applies the framework to algebraic spheres and various fields (finite, local, imaginary number fields, and fields of cohomological dimension $\le 2$), providing both general criteria and explicit constructions (notably Müller's $\mathbb{Q}_2$ example) that demonstrate computability and limitations of the method.
Abstract
We investigate the question of the existence of a non-vanishing section of the tangent bundle on a smooth affine quadric hypersurface $Q^o$ over a given perfect field $k$. In case $Q^o$ admits a $k$-rational point, we give necessary and sufficient conditions for such existence. We apply these conditions in a number of examples, including the case of the algebraic $n$-sphere over $k$, $S^n_k\subset \mathbb{A}^{n+1}_k$, defined by the equation $\sum_{i=1}^{n+1}x_i^2=1$.