Killing spinor-valued forms and their integrability conditions
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Killing spinor‑valued forms and their integrability conditions Petr Somberg1 · Petr Zima1 Received: 4 April 2020 / Accepted: 23 July 2020 © Springer Nature B.V. 2020
Abstract We study invariant systems of PDEs defining Killing vector-valued forms, and then we specialize to Killing spinor-valued forms. We give a detailed treatment of their prolongation and integrability conditions by relating the pointwise values of solutions to the curvature of the underlying manifold. As an example, we completely solve the equations on model spaces of constant curvature producing brand-new solutions which do not come from the tensor product of Killing spinors and Killing–Yano forms. Keywords Killing-type equations · Prolongation of differential systems · Projective invariance · Spinor-valued differential forms · Cone construction · Constant curvature space Mathematics Subject Classification 35N10 · 53A20 · 53A55 · 53B20 · 53C21 · 58J70
1 Introduction Killing equations are a class of invariant overdetermined systems of partial differential equations, appearing naturally in many problems related to (pseudo-)Riemannian geometry. One of the most prominent examples are the Killing vectors, corresponding to infinitesimal isometries of Riemannian manifolds. In the present article, we focus on another specific example in the hierarchy of Killing equations, termed Killing spinor-valued forms. We introduce relevant Killing equations and deduce their properties mostly implied by integrability of the differential system in question. We shall start our analysis in a rather general context and then gradually specialize to the cases of most authors’ interest. As an application of general results, we shall completely resolve the Killing equations on model spaces of constant curvature.
The authors gratefully acknowledge the support of the Grants GACR 306-33/1906357, GAUK 700217 and SVV-2017-260456. * Petr Zima [email protected] Petr Somberg [email protected] 1
Mathematical Institute of Charles University, Sokolovská 83, Praha 8, Karlín, Czech Republic
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Annals of Global Analysis and Geometry
The main motivation for the study of Killing spinor-valued forms is that they are a natural generalization of both Killing spinors and Killing–Yano forms. The Killing spinors and Killing–Yano forms play a dominant role in the geometrical analysis on Riemannian manifolds, e.g., the study of Dirac and Laplace operators and the associated eigenvalue problems. Subsequently, the two examples of Killing-type equations gained their own interest in theoretical physics, too. A central question in the subject asks for (pseudo-)Riemannian manifolds admitting nontrivial solutions of Killing-type equations, and their relation to the underlying geometric structure for which they occur. To some extent, this question is answered by the integrability conditions which relate the solutions with the curvature properties of manifolds. Moreover, the Killing spinors and Killing–Yano forms are closely related to special Riemannian
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