We generalize the spinorial characterization of isometric immersions of surfaces in given by T. Friedrich to surfaces in and . 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 -space.
Mots-clés : spin geometry, surface, energy-momentum tensor
@article{TSG_2004-2005__23__131_0, author = {Morel, Bertrand}, title = {Surfaces in $\mathbb{S}^3$ and $\mathbb{H}^3$ via spinors}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {131--144}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {23}, year = {2004-2005}, doi = {10.5802/tsg.235}, zbl = {1106.53004}, mrnumber = {2270227}, language = {en}, url = {http://www.numdam.org/articles/10.5802/tsg.235/} }
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 SP - 131 EP - 144 VL - 23 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.235/ DO - 10.5802/tsg.235 LA - en ID - TSG_2004-2005__23__131_0 ER -
%0 Journal Article %A Morel, Bertrand %T Surfaces in $\mathbb{S}^3$ and $\mathbb{H}^3$ via spinors %J Séminaire de théorie spectrale et géométrie %D 2004-2005 %P 131-144 %V 23 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.235/ %R 10.5802/tsg.235 %G en %F TSG_2004-2005__23__131_0
Morel, Bertrand. Surfaces in $\mathbb{S}^3$ and $\mathbb{H}^3$ via spinors. Séminaire de théorie spectrale et géométrie, Volume 23 (2004-2005), pp. 131-144. doi : 10.5802/tsg.235. http://www.numdam.org/articles/10.5802/tsg.235/
[1] C. Bär, Real Killing Spinors and Holonomy, Commun. Math. Phys. 154 (1993), 509-521. | MR | Zbl
[2] —,Metrics with Harmonic Spinors, Geom. Func. Anal. 6 (1996), 899–942. | MR | Zbl
[3] C. Bär, P. Gauduchon and A. Moroianu, Generalized Cylinders in Semi-Riemannian and Spin Geometry, math.DG/0303095. | Zbl
[4] H. Baum, Th. Friedrich, R. Grunewald, and I. Kath, Twistor and Killing Spinors on Riemannian Manifolds, Teubner Verlag, Stuttgart/Leipzig, 1991. | MR | Zbl
[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 -space, J. Geom. Phys. 28 (1998), no. 1-2, 143–157. | MR | Zbl
[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. | MR | Zbl
[8] O. Hijazi, Lower bounds for the eigenvalues of the Dirac operator, J. Geom. Phys. 16 (1995), 27–38. | MR | Zbl
[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. | MR | Zbl
[12] B. Morel, Eigenvalue Estimates for the Dirac-Schrödinger Operators, J. Geom. Phys. 38 (2001), 1–18. | MR | Zbl
[13] —, The Energy-Momentum tensor as a second fundamental form, math.DG/0302205.
[14] I.A. Taimanov, The Weierstrass representation of closed surfaces in , Funct. Anal. Appl. 32 (1998), no. 4, 49–62. | MR | Zbl
[15] A. Trautman, Spinors and the Dirac operator on hypersurfaces I. General Theory, Journ. Math. Phys. 33 (1992), 4011–4019. | MR | Zbl
Cited by Sources: