Surfaces in 𝕊 3 and ℍ 3 via spinors

Bertrand Morel

Séminaire de théorie spectrale et géométrie (2004-2005)

  • Volume: 23, page 131-144
  • ISSN: 1624-5458

Abstract

top
We generalize the spinorial characterization of isometric immersions of surfaces in ℝ 3 given by T. Friedrich to surfaces in 𝕊 3 and ℍ 3 . The main argument is the interpretation of the energy-momentum tensor associated with a special spinor field as a second fundamental form. It turns out that such a characterization of isometric immersions in terms of a special section of the spinor bundle also holds in the case of hypersurfaces in the Euclidean 4 -space.

How to cite

top

Morel, Bertrand. "Surfaces in $\mathbb{S}^3$ and $\mathbb{H}^3$ via spinors." Séminaire de théorie spectrale et géométrie 23 (2004-2005): 131-144. <http://eudml.org/doc/11208>.

@article{Morel2004-2005,
abstract = {We generalize the spinorial characterization of isometric immersions of surfaces in $\mathbb\{R\}^3$ given by T. Friedrich to surfaces in $\mathbb\{S\}^3$ and $\mathbb\{H\}^3$. The main argument is the interpretation of the energy-momentum tensor associated with a special spinor field as a second fundamental form. It turns out that such a characterization of isometric immersions in terms of a special section of the spinor bundle also holds in the case of hypersurfaces in the Euclidean $4$-space.},
author = {Morel, Bertrand},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {spin geometry; surface; energy-momentum tensor; hypersurface; isometric immersion; spinor bundle},
language = {eng},
pages = {131-144},
publisher = {Institut Fourier},
title = {Surfaces in $\mathbb\{S\}^3$ and $\mathbb\{H\}^3$ via spinors},
url = {http://eudml.org/doc/11208},
volume = {23},
year = {2004-2005},
}

TY - JOUR
AU - Morel, Bertrand
TI - Surfaces in $\mathbb{S}^3$ and $\mathbb{H}^3$ via spinors
JO - Séminaire de théorie spectrale et géométrie
PY - 2004-2005
PB - Institut Fourier
VL - 23
SP - 131
EP - 144
AB - We generalize the spinorial characterization of isometric immersions of surfaces in $\mathbb{R}^3$ given by T. Friedrich to surfaces in $\mathbb{S}^3$ and $\mathbb{H}^3$. The main argument is the interpretation of the energy-momentum tensor associated with a special spinor field as a second fundamental form. It turns out that such a characterization of isometric immersions in terms of a special section of the spinor bundle also holds in the case of hypersurfaces in the Euclidean $4$-space.
LA - eng
KW - spin geometry; surface; energy-momentum tensor; hypersurface; isometric immersion; spinor bundle
UR - http://eudml.org/doc/11208
ER -

References

top
  1. C. Bär, Real Killing Spinors and Holonomy, Commun. Math. Phys. 154 (1993), 509-521. Zbl0778.53037MR1224089
  2. —,Metrics with Harmonic Spinors, Geom. Func. Anal. 6 (1996), 899–942. Zbl0867.53037MR1421872
  3. C. Bär, P. Gauduchon and A. Moroianu, Generalized Cylinders in Semi-Riemannian and Spin Geometry, math.DG/0303095. Zbl1068.53030
  4. H. Baum, Th. Friedrich, R. Grunewald, and I. Kath, Twistor and Killing Spinors on Riemannian Manifolds, Teubner Verlag, Stuttgart/Leipzig, 1991. Zbl0734.53003MR1164864
  5. J.P. Bourguignon, O. Hijazi, J.-L. Milhorat, and A. Moroianu, A Spinorial approach to Riemannian and Conformal Geometry, Monograph, In preparation, 2004. 
  6. Th. Friedrich, On the spinor representation of surfaces in Euclidean 3 -space, J. Geom. Phys. 28 (1998), no. 1-2, 143–157. Zbl0966.53042MR1653146
  7. Th. Friedrich and E.-C. Kim, The Einstein-Dirac equation on Riemannian spin manifolds, J. Geom. Phys. 33 (2000), no. 1–2, 128–172. Zbl0961.53023MR1738150
  8. O. Hijazi, Lower bounds for the eigenvalues of the Dirac operator, J. Geom. Phys. 16 (1995), 27–38. Zbl0823.34074MR1325161
  9. R. Kusner and N. Schmitt, The Spinor Representation of Surfaces in Space, math.DG-GA/9610005. 
  10. —, The Spinor Representation of Minimal Surfaces, math.DG-GA/9512003 
  11. H.B. Lawson and M.L. Michelsohn, Spin Geometry, Princeton Univ. Press, 1989. Zbl0688.57001MR1031992
  12. B. Morel, Eigenvalue Estimates for the Dirac-Schrödinger Operators, J. Geom. Phys. 38 (2001), 1–18. Zbl0984.53021MR1817510
  13. —, The Energy-Momentum tensor as a second fundamental form, math.DG/0302205. 
  14. I.A. Taimanov, The Weierstrass representation of closed surfaces in ℝ 3 , Funct. Anal. Appl. 32 (1998), no. 4, 49–62. Zbl0979.53012MR1678856
  15. A. Trautman, Spinors and the Dirac operator on hypersurfaces I. General Theory, Journ. Math. Phys. 33 (1992), 4011–4019. Zbl0769.58055MR1191759

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.