Classification of Flat Lagrangian H-Umbilical Submanifolds In Para-Kähler n-Plane

Year 2011, Volume: 4 Issue: 1, 1 - 14, 30.04.2011

Bang-yen Chen

Abstract

Keywords

Para-Kähler manifold, Lagrangian H-umbilical submanifold

References

• [1] Chen, B.-Y., Geometry of submanifolds and Its applications. Science University of Tokyo, Tokyo, 1981.
• [2] Chen, B.-Y., Total mean curvature and submanifolds of finite type, World Scientific, New Jersey, 1984.
• [3] Chen, B.-Y., Interaction of Legendre curves and Lagrangian submanifolds, Israel J. Math. 99 (1997), 69–108.
• [4] Chen, B.-Y., Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tˆohoku Math. J., 49 (1997), no. 2, 277–297.
• [5] Chen, B.-Y., Representation of flat Lagrangian H-umbilical submanifolds in complex Eu- clidean spaces, Tohoku Math. J., 51 (1999), no. 1, 13–20.
• [6] Chen, B.-Y., Riemannian submanifolds, in: Handbook of Differential Geometry, Vol. I, 187– 418, North-Holland, Amsterdam, (eds. F. Dillen and L. Verstraelen), 2000.
• [7] Chen, B.-Y., Riemannian geometry of Lagrangian submanifolds, Taiwanese J. Math., 5 (2001), no. 4, 681–723.
• [8] Chen, B.-Y., Lagrangian submanifolds in para-K¨ahler manifolds, Nonlinear Analysis, 73 (2010), no. 11, 3561–3571.
• [9] Chen, B.-Y., Lagrangian H-umbilical submanifolds of para-K¨ahler manifolds, Taiwanese J. Math., (to appear).
• [10] Chen, B.-Y. and Ogiue, K., On totally real submanifolds, Trans. Amer. Math. Soc., 193 (1974), 257–266.
• [11] Cort´es, V., The special geometry of Euclidean supersymmetry: a survey, Rev. Un. Mat. Argentina, 47 (2006), no. 1, 29–34.
• [12] Cort´es, V., Lawn, M.-A. and Sch¨afer, L., Affine hyperspheres associated to special para- K¨ahler manifolds, Int. J. Geom. Methods Mod. Phys., 3 (2006), no. 5–6, 995–1009.
• [13] Cort´es, V., Mayer, C., Mohaupt, T. and Saueressig, F., Special geometry of Euclidean super- symmetry. I, Vector multiplets, J. High Energy Phys., 2004, no. 3, 028, 73 pp.
• [14] Etayo, F., Santamar´ıa, R. and Tr´ıas, U. R., The geometry of a bi-Lagrangian manifold, Differential Geom. Appl., 24 (2006), no. 1, 33–59.
• [15] Hou, Z., Deng, S. and Kaneyuki, S., Dipolarizations in compact Lie algebras and homogeneous para-Ka¨hler manifold, Tokyo J. Math., 20 (1997), no. 2, 381–388.
• [16] Ponge, R. and Reckziegel, H., Twisted products in pseudo-Riemannian geometry, Geometriae Dedicata, 48 (1993), no. 1, 15–25.
• [17] Rashevskij, P. K., The scalar field in a stratified space, Trudy Sem. Vektor. Tenzor. Anal., 6 (1948), 225–248.
• [18] Rozenfeld, B. A., On unitary and stratified spaces, Trudy Sem. Vektor. Tenzor. Anal., 7 (1949), 260–275.
• [19] Ruse, H. S., On parallel fields of planes in a Riemannian manifold, Quart. J. Math. Oxford Ser., 20 (1949), 218–234.