The concept of a “spinor” emerged from the work of E. Cartan on the representations of simple Lie algebras. However, it was not until Dirac employed a special case in the construction of his relativistically invariant equation for the electron with “spin” that the notion acquired its present name or its current stature in mathematical physics. In this chapter we present an elementary introduction to the algebraic theory of spinors in Minkowski spacetime and illustrate its utility in special relativity by recasting in spinor form much of what we have learned about the structure of the electromagnetic field in Chapter 2. We shall not stray into quantum mechanics and, in particular, will not discuss the Dirac equation (for this, see the encyclopedic monograph[PR] of Penrose and Rindler). Since it is our belief that an intuitive appreciation of the notion of a spinor is best acquired by approaching them by way of group representations, we have devoted this first section to an introduction to these ideas and how they arise in special relativity. Since this section is primarily motivational, we have not felt compelled to prove everything we say and have, at several points, contented ourselves with a reference to a proof in the literature.

The Theory of Spinors

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Annals of Global Analysis and Geometry

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel-Lizorkin spaces on manifolds, improving [11].

Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel-Lizorkin spaces on manifolds, improving a result by Skrzypczak.

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contain local elliptic boundary conditions in the sense of Lopatinski and Shapiro as well as the Atiyah–Patodi–Singer boundary conditions. We discuss boundary regularity of solutions and also spectral and index theory. The emphasis is on providing the reader with a working knowledge.

/pdf/guide-to-elliptic-boundary-value-problems-for-dirac-type-3rqu3l9wzn.pdf

Guide to Elliptic Boundary Value Problems for Dirac-Type Operators

Simply connected three-dimensional homogeneous manifolds ${\mathbb{E}(\kappa, \tau)}$, with four-dimensional isometry group, have a canonical Spinc structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into ${\mathbb{E}(\kappa, \tau)}$. As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in ${\mathbb{E}(\kappa, \tau)}$. Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spinc spinors.

Hypersurfaces of Spin c Manifolds and Lawson Type Correspondence

On Spin^c 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 Spin^c manifold. Using the notion of generalized cylinders, we derive the variationnal 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 Spin^c Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.

https://hal.archives-ouvertes.fr/hal-00492141/document

The Energy-Momentum Tensor on $Spin^c$ Manifolds

We extend to the eigenvalues of the hypersurface Spinc Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Furthermore, we prove a correspondence between the existence of a Spinc Killing spinor on homogeneous 3-dimensional manifolds E ⁎ ( κ , τ ) with 4-dimensional isometry group and isometric immersions of E ⁎ ( κ , τ ) into the complex space form M 4 ( c ) of constant holomorphic sectional curvature 4c, for some c ∈ R ⁎ . As applications, we show the non-existence of totally umbilic surfaces in E ⁎ ( κ , τ ) and we give necessary and sufficient geometric conditions to immerse a 3-dimensional Sasaki manifold into M 4 ( c ) .

https://pure.mpg.de/pubman/item/item_3121519_1/component/file_3121520/Nakad_the%20spin_oa_2013.pdf

The Spinc Dirac operator on hypersurfaces and applications

On $Spin^c$ 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 $Spin^c$ manifold. Using the notion of generalized cylinders, we derive the variationnal 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 $Spin^c$ Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.

The Energy-Momentum tensor on $Spin^c$ manifolds

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. B\"ar and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound of Hijazi-Montiel-Zhang, involving the mean curvature of the boundary, for the spectrum of the Dirac operator on the noncompact boundary of a Spin$^c$ manifold. The limiting case is then studied and examples are then given.

Roger Nakad

Papers

Hypersurfaces of Spin c Manifolds and Lawson Type Correspondence

The Energy-Momentum Tensor on $Spin^c$ Manifolds

The Spinc Dirac operator on hypersurfaces and applications

The Energy-Momentum tensor on $Spin^c$ manifolds

Boundary value problems for noncompact boundaries of Spin$^c$ manifolds and spectral estimates