The length function of a twistor spinor with zero
Let \((M^n , g)\), be an \(n\)-dimensional Riemannian spin manifold and \(\Sigma M\) its spinor bundle. Let \(\nabla^\Sigma\) and \(D\) denote the spinor derivative and the Dirac operator on \(M^n\) respectively. A spinor field \(\Phi\) is called a twistor spinor if it satisfies the equation \(\nabla^\Sigma_X\Phi + \frac1n X.D \Phi = 0\), for all vectors fields \(X\) on \(M\), where \(.\) denotes the Clifford multiplication defined by the spinor representation. Such an equation is called the twistor equation. The paper under review is a presentation of important facts concerning the Taylor expansion of the length function of a twistor spinor with zero on a Riemannian orbifold around its zero. The author gives a thorough study on the Eguchi-Hanson orbifold. The expansion is written by using some conformal normal coordinates, briefly called \((CNC)\), around the zero up to order 7. In the example of the Eguchi-Hanson orbifold, the conformal normal coordinates \((CNC)\) are given explicitly. The author uses the expansion for computing the mass, a generalization of \((ADM)\)-mass, of the asymptotically locally Euclidean coordinate system, which is constructed from a conformal normal coordinate system around the zero of a twistor spinor on a Riemannian spin orbifold admitting isolated singularities.
scientific article; zbMATH DE number 6018064
scientific article