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We develop a new framework for the study of generalized Killing spinors, where every generalized Killing spinor equation, possibly with constraints, can be formulated equivalently as a system of partial differential equations for a polyform satisfying algebraic relations in the Kähler–Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Σ of real type as a real algebraic variety in the Kähler–Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS 4 space-time. [ABSTRACT FROM AUTHOR]

In this paper, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost 3-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the 3-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact (qc) structure of the quaternionic Heisenberg groups. We focus on the 7-dimensional case and prove that the canonical connection can also be obtained by means of a cocalibrated G2 structure. We then study the spinorial properties of this group and present the noteworthy fact that it is the only known example of a manifold which carries generalized Killing spinors with three different eigenvalues. [ABSTRACT FROM AUTHOR]

We study generalized Killing spinors on the standard sphere S 3 , which turn out to be related to Lagrangian embeddings in the nearly Kähler manifold S 3 × S 3 and to great circle flows on S 3 . Using our methods we generalize a well known result of Gluck and Gu [5] concerning divergence-free geodesic vector fields on the sphere and we show that the space of Lagrangian submanifolds of S 3 × S 3 has at least three connected components. [ABSTRACT FROM AUTHOR]

We study generalized Killing spinors on round spheres $$\mathbb {S}^n$$ . We show that on the standard sphere $$\mathbb {S}^8$$ any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on $$\mathbb {S}^n$$ whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola-Friedrich's canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors. [ABSTRACT FROM AUTHOR]

We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence of compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi-Yau, hyperkähler, G2 and Spin(7) manifolds. [ABSTRACT FROM AUTHOR]

On Spinc manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a Spinc manifold. Using the notion of generalized cylinders, we derive the variational formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized Spinc Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors. [ABSTRACT FROM AUTHOR]

Let M0= G0/ H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition $\mathfrak {g}_{0}=\mathfrak {h}+\mathfrak {m}$ and let S( M0) be the spin bundle defined by the spin representation $\tilde{ \operatorname {Ad}}:H\rightarrow \mathrm {GL}_{\mathbb {R}}(S)$ of the stabilizer H. This article studies the superizations of M0, i.e. its extensions to a homogeneous supermanifold M= G/ H whose sheaf of superfunctions is isomorphic to the sheaf of sections of Λ( S*( M0)). Here G is the Lie supergroup associated with a certain extension of the Lie algebra of symmetry $\mathfrak {g}_{0}$ to an algebra of supersymmetry $\mathfrak {g}=\mathfrak {g}_{\overline {0}}+\mathfrak {g}_{\overline {1}}=\mathfrak {g}_{0}+S$ via the Kostant-Koszul construction. Each algebra of supersymmetry naturally determines a flat connection $\nabla^{\mathcal {S}}$ in the spin bundle S( M0). Killing vectors together with generalized Killing spinors (i.e. $\nabla^{\mathcal {S}}$ -parallel spinors) are interpreted as the values of appropriate geometric symmetries of M, namely even and odd Killing fields. An explicit formula for the Killing representation of the algebra of supersymmetry is obtained, generalizing some results of Koszul. The generalized spin connection $\nabla^{\mathcal {S}}$ defines a superconnection on M, via the super-version of a theorem of Wang. [ABSTRACT FROM AUTHOR]

Abstract: We give a spinorial characterization of isometrically immersed hypersurfaces into 4-dimensional space forms and product spaces , in terms of the existence of particular spinor fields, called generalized Killing spinors or equivalently solutions of a Dirac equation. This generalizes to higher dimensions several recent results on the spinorial Weierstraß representation by U. Abresch, D. Sullivan, R. Kusner, N. Schmidt and many others. The main argument is the interpretation of the energy–momentum tensor of a generalized Killing spinor as the second fundamental form, possibly up to a tensor depending on the ambient space. As an application, we deduce some non-existence results for isometric immersions into the 4-dimensional Euclidean space. [Copyright &y& Elsevier]