THE CROSS-RATIO MANIFOLD: A MODEL OF CENTRO-AFFINE GEOMETRY

Yıl 2011, Cilt: 4 Sayı: 2, 32 - 62, 30.10.2011

Roland Hildebrand

Öz

Anahtar Kelimeler

para-Kähler manifolds, Lagrangian immersions, Codazzi structures

Kaynakça

• [1] Alekseevsky, D.V., Medori, C. and Tomassini, A., Homogeneous para-Kähler Einstein manifolds, Russ. Math. Surv., 64(2009), no. 1, 1–43.
• [2] Ariyawansa, K.A., Davidon, W.C. and McKennon, K.D., A coordinate-free foundation for projective spaces treating projective maps from a subset of a vector space into another vector space, Tech. Rep. 2004-4, Washington State Univ., 2004.
• [3] Calabi, E., Hypersurfaces with maximal affinely invariant area, Amer. J. Math., 104(1982), no. 1, 91-126.
• [4] Chen, B.-Y., Lagrangian submanifolds in para-Kähler manifolds, Nonlinear Analysis, 73(2010), no. 11, 3561–3571.
• [5] Chursin, M., Sch¨afer, L. and Smoczyk, K., Mean curvature flow of space-like Lagrangian submanifolds in almost para-K¨ahler manifolds, Calc. Var., 41(2010), no. 1–2, 111–125.
• [6] Cortés, V., Mayer, C., Mohaupt, T. and Saueressig, F., Special geometry of Euclidean super- symmetry I: Vector multiplets, J. High Energy Phys., 03(2004), 028.
• [7] Cruceanu, V., Fortuny, P. and Gadea, P.M., A survey on paracomplex geometry, Rocky Mountain J. Math., 26(1996), no. 1, 83–115.
• [8] Etayo, F., Santamaría, R. and Trías, U.R., The geometry of a bi-Lagrangian manifold, Differ. Geom. Appl., 24(2006), no. 1, 33–59.
• [9] Gadea, P.M. and Montesinos Amilibia, A., The paracomplex projective spaces as symmetric and natural spaces, Indian J. Pure Appl. Math., 23(1992), no. 4, 261–275.
• [10] Gadea, P.M. and Montesinos Amilibia, A., Spaces of constant para-holomorphic sectional curvature, Pac. J. Math., 136(1989), no. 1, 85–101.
• [11] Hildebrand, R., Half-dimensional immersions in para-K¨ahler manifolds, Int. Electron. J. Geom., 4(2011), no.2, 85-113.
• [12] Kobayashi, S. and Nomizu, K., Foundations of differential geometry II, Interscience publish- ers, New York, London, Sydney, 1969.
• [13] Libermann, P., Sur le probl`eme d’équivalence de certaines structures infinit´esimales, Ann. Mat. Pura Appl., 36(1954), 27–120.
• [14] Loftin, J.C., Affine spheres and Kähler-Einstein metrics, Math. Res. Lett., 9(2002), no. 4, 425–432.
• [15] Nomizu, K. and Sasaki, T., Affine differential geometry: geometry of affine immersions, Vol. 111 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1994.
• [16] Shima, H., Geometry of Hessian Structures, World Scientific, Hackensack, New Jersey, 2007.
• [17] Simon, U., A survey on Codazzi tensors, In memoriam Ernst Mohr, Mathematiker und Opfer des Faschismus, Vol. 8 of Forum der Berliner Math. Gesellschaft, 97-106, Berliner Math. Gesellschaft, Berlin, 2009.
• [18] Vaisman, I., Symplectic curvature tensors, Monatshefte fu¨r Mathematik, 100(1985), no. 4, 299-327.
• [19] Vrancken, L., Centroaffine differential geometry and its relations to horizontal submanifolds, PDEs, submanifolds and affine differential geometry, Vol. 57 of Banach Center Publ., 21-28, Polish Acad. Sci., Warsaw, 2002.