Editing Dirac operator (section)

=== Example 5 ===
For a [[spin manifold]], ''M'', the Atiyah–Singer–Dirac operator is locally defined as follows: For {{nowrap|''x'' ∈ ''M''}} and ''e<sub>1</sub>''(''x''), ..., ''e<sub>j</sub>''(''x'') a local orthonormal basis for the tangent space of ''M'' at ''x'', the Atiyah–Singer–Dirac operator is
:<math>D=\sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)} ,</math>
where <math>\tilde{\Gamma}</math> is the [[spin connection]], a lifting of the [[Levi-Civita connection]] on ''M'' to the [[spinor bundle]] over ''M''.  The square in this case is not the Laplacian, but instead <math>D^2=\Delta+R/4</math> where <math>R</math> is the [[scalar curvature]] of the connection.<ref> Jurgen Jost, (2002) "Riemannian Geometry ang Geometric Analysis (3rd edition)", Springer. ''See section 3.4 pages 142 ff.''</ref>