Skew Killing spinors in four dimensions

Nicolas Ginoux (1) , Georges Habib (2) , Ines Kath (3)

1 IECL - Institut Élie Cartan de Lorraine
2 LU / ULB - الجامعة اللبنانية [بيروت] = Lebanese University [Beirut] = Université libanaise [Beyrouth]
3 Institut für Mathematik und Informatik

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor ψ is a spinor that satisfies the equation ∇Xψ = AX · ψ with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product R × N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to S 2 × R 2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.

Mots clés: Hodge operator; doubly warped product; Generalized Killing spinors

Domaines: Géométrie différentielle [math.DG]

Nicolas Ginoux, Georges Habib, Ines Kath. Skew Killing spinors in four dimensions. Annals of Global Analysis and Geometry, 2021, 59 (4), pp.501-535. ⟨10.1007/s10455-021-09754-9⟩. ⟨hal-02560712v2⟩