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Hypersurfaces of Spinc Manifolds and Lawson Type Correspondence

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Niveau: Supérieur, Doctorat, Bac+8
Hypersurfaces of Spinc Manifolds and Lawson Type Correspondence Roger Nakad and Julien Roth March 24, 2012 Abstract Simply connected 3-dimensional homogeneous manifoldsE(?, ?), with 4-dimen- sional 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 E(?, ?). As applica- tion, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in E(?, ?). Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spinc spinors. Keywords: Spinc structures, Killing and parallel spinors, isometric immersions, Law- son type correspondence, Sasaki hypersurfaces. Mathematics subject classifications (2010): 58C40, 53C27, 53C40, 53C80. 1 Introduction It is well-known that a conformal immersion of a surface in R3 could be characterized by a spinor field ? satisfying D? = H?, (1) where D is the Dirac operator and H the mean curvature of the surface (see [12] for instance). In [4], Friedrich characterized surfaces in R3 in a geometrically invariant way. More precisely, consider an isometric immersion of a surface (M2, g) into R3.

  • killing constant

  • dimensional homogeneous

  • riemannian surface

  • spinor bundle

  • constant curvature

  • i?

  • sider hypersurfaces

  • mean curvature

  • characterized via spinc spinors


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Language English
Hypersurfaces of SpincManifolds and Lawson Type Correspondence
Roger Nakad and Julien Roth
May 24, 2012
Abstract
Simply connected3-dimensional homogeneous manifoldsE(κ, τ), with4-dimen-sional isometry group, have a canonicalSpincstructure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces intoE(κ, τ). As applica-tion, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces inE(κ, τ). Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized viaSpincspinors. Keywords:Spincstructures, Killing and parallel spinors, isometric immersions, Law-son type correspondence, Sasaki hypersurfaces.
Mathematics subject classifications (2010):58C40, 53C27, 53C40, 53C80.
1 Introduction
It is well-known that a conformal immersion of a surface inR3could be characterized by a spinor fieldϕsatisfying =Hϕ,(1)
whereDis the Dirac operator andHthe mean curvature of the surface (see [12] for instance). In [4], Friedrich characterized surfaces inR3in a geometrically invariant way. More precisely, consider an isometric immersion of a surface(M2, g)intoR3. The restriction toMof a parallel spinor ofR3satisfies, for allXΓ(T M), the following relation rXϕ=12IIXϕ,(2) whereris the spinorial Levi-Civita connection ofM, “” denotes the Clifford multi-plication ofMandII Hence,is the shape operator of the immersion.ϕis a solution
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of the Dirac equation (1) with constant norm. Conversely, assume that a Riemannian surface(M2, g)carries a spinor fieldϕ, satisfying rXϕ=12EXϕ,(3) whereE It is straightfor-is a given symmetric endomorphism on the tangent bundle. ward to see thatE= 2`ϕ.Here`ϕis a field of symmetric endomorphisms associated with the field of quadratic forms, denoted also by`ϕ, called the energy-momentum tensor which is given, on the complement set of zeroes ofϕ, by `ϕ(X) =<X• rXϕ,|ϕϕ|2, for anyXΓ(T M) the existence of a pair. Then,(ϕ, E)satisfying (3) implies that the tensorE= 2`ϕsatisfies the Gauss and Codazzi equations and by Bonnet’s the-orem, there exists a local isometric immersion of(M2, g)intoR3withEas shape operator. Friedrich’s result was extended by Morel [16] for surfaces of the sphereS3 and the hyperbolic spaceH3.
Recently, the second author [25] gave a spinorial characterization of surfaces iso-metrically immersed into3-dimensional homogeneous manifolds with4-dimensional isometry group. These manifolds, denoted byE(κ, τ)are Riemannian fibrations over a simply connected 2-dimensional manifoldM2(κ)with constant curvatureκand bun-dle curvatureτ. This fibration can be represented by a unit vector fieldξtangent to the fibers.
The manifoldsE(κ, τ)areSpinhaving a special spinor fieldψ. This spinor is constructed using real or imaginary Killing spinors onM2(κ). Ifτ6= 0, the restriction ofψto a surface gives rise to a spinor fieldϕsatisfying, for every vector fieldX, rXϕ1=IIXϕ+iτ2Xϕiα2g(X, T)Tϕ+i2αgf(X, T)ϕ.(4) 2 Hereα= 2τ2κτ,fis a real function andTis a vector field onMsuch thatξ=T+f ν is the decomposition ofξinto tangential and normal parts (νis the normal vector field of the immersion). The spinorϕis given byϕ:=ϕ+ϕ, whereϕ=ϕ++ϕis the decomposition into positive and negative spinors. Up to some additional geomet-ric assumptions onTanff, the spinorϕallows to characterize the immersion of the surface intoE(κ, τ)[25].
In the present paper, we considerSpincstructures onE(κ, τ)instead ofSpinstruc-tures. The manifoldsE(κ, τ)have a canonicalSpincstructure carrying a natural spinor field, namely a real Killing spinor with Killing constantτ2 restriction of this. The Killing spinor toMgives rise to a special spinor satisfying τ rXϕ=12IIXϕ+i2Xϕ.
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