Editing Dirac operator (section)

=== Example 6 ===
On [[Riemannian manifold]] <math>(M, g)</math> of dimension <math>n=dim(M)</math> with [[Levi-Civita connection]] <math>\nabla</math>and an [[orthonormal basis]] <math>\{e_{a}\}_{a=1}^{n}</math>, we can define [[exterior derivative]] <math>d</math> and [[Codifferential|coderivative]] <math>\delta</math> as
: <math>d= e^{a}\wedge \nabla_{e_{a}}, \quad \delta =e^{a} \lrcorner \nabla_{e_{a}}</math>.

Then we can define a Dirac-Kähler operator<ref name=":0">{{Cite journal |last=Graf |first=Wolfgang |date=1978 |title=Differential forms as spinors |url=http://www.numdam.org/item/?id=AIHPA_1978__29_1_85_0 |journal=Annales de l'Institut Henri Poincaré A |language=en |volume=29 |issue=1 |pages=85–109 |issn=2400-4863}}</ref><ref name=":1">{{Cite book |last1=Benn |first1=Ian M. |url=https://books.google.com/books?id=FzcbAQAAIAAJ |title=An Introduction to Spinors and Geometry with Applications in Physics |last2=Tucker |first2=Robin W. |date=1987 |publisher=A. Hilger |isbn=978-0-85274-169-6 |language=en}}</ref><ref name=":2">{{Cite journal |last=Kycia |first=Radosław Antoni |date=2022-07-29 |title=The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics |url=https://doi.org/10.1007/s00025-022-01646-z |journal=Results in Mathematics |language=en |volume=77 |issue=5 |pages=182 |doi=10.1007/s00025-022-01646-z |arxiv=2009.08542 |s2cid=221802588 |issn=1420-9012}}</ref> <math>D</math>, as follows
: <math>D = e^{a}\nabla_{e_{a}}=d-\delta</math>.

The operator acts on sections of [[Clifford bundle]] in general, and it can be restricted to spinor bundle, an ideal of Clifford bundle, only if the projection operator on the ideal is parallel.<ref name=":0" /><ref name=":1" /><ref name=":2" />