Twistor Geometry of Null Foliations in Complex Euclidean Space

Abstract

We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface ${\mathcal{Q}}^n$ of dimension $n\ge 3$, and its twistor space $\mathbb{PT}$, defined to be the space of all linear subspaces of maximal dimension of ${\mathcal{Q}}^n$. Viewing complex Euclidean space ${\mathbb{CE}}^n$ as a dense open subset of ${\mathcal{Q}}^n$, we show how local foliations tangent to certain integrable holomorphic totally null distributions of maximal rank on ${\mathbb{CE}}^n$ can be constructed in terms of complex submanifolds of $\mathbb{PT}$. The construction is illustrated by means of two examples, one involving conformal Killing spinors, the other, conformal Killing-Yano $2$-forms. We focus on the odd-dimensional case, and we treat the even-dimensional case only tangentially for comparison.

References in corpus (1)

• The conformal Killing equation on forms -- prolongations and applications

Cited by in corpus (3)

• Twistor theory at fifty: from contour integrals to twistor strings
• A Projective-to-Conformal Fefferman-Type Construction
• Almost Robinson geometries