Editing Killing spinor (section)

== Definition ==
By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those [[Twistor theory|twistor]]
spinors which are also [[eigenspinor]]s of the [[Dirac operator]].<ref>{{cite journal|title=Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkei nichtnegativer Skalarkrümmung|author=Th. Friedrich|journal=[[Mathematische Nachrichten]]|volume=97|year=1980|pages=117–146|doi=10.1002/mana.19800970111}}</ref><ref>{{cite journal|title=On the conformal relation between twistors and Killing spinors|author=Th. Friedrich|journal=Supplemento dei Rendiconti del Circolo Matematico di Palermo, Serie II|volume=22|year=1989|pages=59–75}}</ref><ref>{{cite journal|title=Spin manifolds, Killing spinors and the universality of Hijazi inequality|author=A. Lichnerowicz|author-link=André Lichnerowicz|journal=Lett. Math. Phys.|volume=13|year=1987|issue=4 |pages=331–334|doi=10.1007/bf00401162|bibcode = 1987LMaPh..13..331L |s2cid=121971999}}</ref> The term is named after [[Wilhelm Killing]].

Another equivalent definition is that Killing spinors are the solutions to the [[Killing equation]] for a so-called Killing number.

More formally:<ref>{{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] |pages= 116–117| year=2000|isbn=978-0-8218-2055-1}}
</ref>

:A '''Killing spinor''' on a  [[Riemannian manifold|Riemannian]] [[Spin structure|spin]] [[manifold]] ''M'' is a [[spinor field]] <math>\psi</math> which satisfies

::<math>\nabla_X\psi=\lambda X\cdot\psi</math>

:for all [[tangent space|tangent vectors]] ''X'', where <math>\nabla</math> is the spinor [[covariant derivative]], <math>\cdot</math> is [[Clifford multiplication]] and <math>\lambda \in \mathbb{C}</math> is a constant, called the '''Killing number''' of <math>\psi</math>. If <math>\lambda=0</math> then the spinor is called a parallel spinor.