Combing a hedgehog over a field

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 $k$, $S_k^n\subset {\mathbb{A}}_k^{n+1}$, defined by the equation ${\sum }_{i=1}^{n+1}{x}_i^2=1$.

arXiv commentadded to the introduction information about the origin and atribution of the main question. Added a construction of an explicit section over the 2-adics by Peter Müller. This is the final version

References in corpus (6)

• An exact sequence for Milnor's K-theory with applications to quadratic forms
• A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
• Fundamental classes in motivic homotopy theory
• Slices of hermitian K-theory and Milnor's conjecture on quadratic forms
• Affine representability results in ${\mathbb{A}}^1$-homotopy theory I: vector bundles
• Motivic Gauß-Bonnet formulas