we get with corollary 7 that i ? P(E i , J i ) Since ? P a (E, J E ) the (0, 2)?curvature of vanishes, With equation (4.2.4) we get Bibliography [Ab] U. Abresch, Spinor Representation of CMC surfaces, Lectures at Luminy, 1989. ,
Classification of N-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of Spin, Commun. Math. Phys, pp.183-477, 1997. ,
Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. Lond. A, pp.362-425, 1978. ,
A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions, Results in Mathematics, vol.55, issue.1-2, pp.3-24, 2003. ,
DOI : 10.1007/BF03322907
Extrinsic Bounds for Eigenvalues of the Dirac Operator, Annals of Global Analysis and Geometry, vol.16, issue.6, pp.573-596, 1998. ,
DOI : 10.1023/A:1006550532236
The existence problem of hyperbolic structures on vector bundles, Publ. Inst. Math. (Beograd), pp.53-67, 1993. ,
Generalized cylinders in semi-riemannian and spin geometry, Math. Z, vol.249, pp.545-580, 2005. ,
Möbius Invariant Flows of Tori in S 4 , Dissertation an der Technischen, 2003. ,
A Survey on Paracomplex Geometry, Rocky Mountain Journal of Mathematics, vol.26, issue.1, pp.83-115, 1996. ,
DOI : 10.1216/rmjm/1181072105
Affine hypersphere associated to special para- Kähler manifolds, to appear in Int, J. Geom. Meth. in Mod. Phys ,
Saueressig Special Geometry of Euclidean Supersymmetry I: Vectormultiplets, J. High Energy Phys, JHEP03, vol.028, p.312001, 2004. ,
The affine differential geometry of complex hypersurfaces, Med. Konink. Acad. Wetensch. Belgie, pp.52-89, 1990. ,
Complex affine hypersurfaces of C n+1, Bull. Soc. Math. Belg. Sér. B, issue.1, pp.40-245, 1988. ,
Eisenhart A treatise on the differential geometry of Curves and Surfaces, 1909. ,
Erdem Harmonic Maps of Lorentz Surfaces, Quadratic Differential and Paraholomorphicity , Beiträge zur Algebra and Geometry, pp.19-32, 1997. ,
Paraholomorphic structures and the connections of vector bundles over paracomplex manifolds, New Zealand J. of Math, vol.30, issue.1, pp.41-50, 2001. ,
Quaternionic holomophic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates or harmonic 2-tori, Invent. Math, pp.146-507, 2001. ,
On the Spinor Representation of Surfaces in Euclidean 3-Space, J. Geom. Phys, vol.28, issue.1, pp.143-157, 1998. ,
Dirac-operatoren in der Riemannschen Geometrie, 1997. ,
Lectures on 2-D gravity and 2-D string theory, preprint hep-th, p.9304011, 1993. ,
The paraquaternionic projective space, 1993. ,
The Fundamental Theorems for Affine Immersionsinto Hyperquadrics and its Applications, Monatshefte f???r Mathematik, vol.131, issue.1, pp.37-48, 2000. ,
DOI : 10.1007/s006050070023
Timelike Minimal Surfaces via Loop Groups, Acta Applicandae Mathematicae, vol.83, issue.3, pp.313-355, 2004. ,
Differential Geometry of Complex Vector Bundles, 1987. ,
DOI : 10.1515/9781400858682
Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann, vol.245, pp.89-99, 1979. ,
Foundations of differential geometry, volume II, 1969. ,
A Weierstraß representation theorem for Lorentz surfaces, Complex Variables, pp.319-112, 2005. ,
Induced surfaces and their integrable dynamics, Stud. Appl. Math, p.96, 1996. ,
The spinor representation of surfaces in space, preprint arXiv:dg-ga, p.1, 1996. ,
Spin geometry, 1989. ,
Decomposition of para-complex vector bundles and paracomplex affine immersions ,
Spin structures on manifolds, Enseign. Math, vol.9, pp.198-203, 1963. ,
Surfaces in S 3 and H 3 via spinors, preprint arXiv: math, p.204090, 2002. ,
On the geometry of affine K??hler immersions, Nagoya Mathematical Journal, vol.41, pp.205-222, 1990. ,
DOI : 10.1007/BF03323250
On the fundamental theorems for purely real immersions, 1999. ,
Semi-Riemannian Geometry with applications to Relativity, 1983. ,
Algebraic curves and non rigid minimal surfaces in the Euclidean space, Pacific Journal of Mathematics, vol.183, issue.2, 1998. ,
DOI : 10.2140/pjm.1998.183.333
Quaternionic analysis on Riemannian surfaces and differential geometry, Doc. Math. J. DMV, Extra, vol.ICM II, pp.389-400, 1998. ,
The conformal Gauss map of submanifolds of the M???bius space, Annals of Global Analysis and Geometry, vol.43, issue.2, pp.97-116, 1987. ,
DOI : 10.1007/BF00127853
<mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>t</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mo>???</mml:mo></mml:msup></mml:math>-bundles in para-complex geometry, special para-K??hler manifolds and para-pluriharmonic maps, Differential Geometry and its Applications, vol.24, issue.1, pp.60-89, 2006. ,
DOI : 10.1016/j.difgeo.2005.07.001
The Weierstrass representation of closed surfaces in R 3, Funct. Anal. Appl, vol.32, issue.4, pp.49-62, 1998. ,
Para-Quaternionic Reduction, preprint, arXiv: math, p.304424, 2003. ,
Foundations of differentiable manifolds and Lie groups, 1971. ,
UntersuchungenüberUntersuchungen¨Untersuchungenüber die Flächen, deren mittlere KrümmungKrümmung¨Krümmungüberall gleich Null ist, Monatsber. Akad. Wiss. Berlin, pp.612-625, 1866. ,
An introduction to Lorentz surfaces, De Gruyter expositions in math, 1996. ,
A note on modified Veselov-Novikov hierarchy, Physics Letters B, vol.454, issue.1-2, pp.454-485, 1999. ,
DOI : 10.1016/S0370-2693(99)00342-1
Lawn Vorname: Marie-Amélie Geburtsdatum: 17.01, 1980. ,
Avenue du Maréchal Juin F-54000 Nancy France Clemens-August-Strasse 76 App. 71 D-53115 Bonn Deutschland email: pailluss@iecn.u-nancy.fr Schullaufbahn 09.1986-06.1990: Grundschule in Vesoul (Frankreich), p.6, 1997. ,