On Isometric Immersions of Null Manifolds into Semi-Riemannian Space Forms of Arbitrary Index

Year 2023, Volume: 16 Issue: 1, 304 - 333, 30.04.2023

Carlos Avila , Matias Navarro , Oscar Palmas , Didier Solis

https://doi.org/10.36890/iejg.1274307

Abstract

A null manifold is a differentiable manifold M endowed with a degenerate metric tensor g. In this work we provide sufficient conditions for a null manifold to be isometrically immersed as a hypersurface into a simple connected semi-Riemannian manifold of constant sectional curvature c and index q

Keywords

Degenerate metric, isometric immersion, Gauss-Codazzi-Ricci equations

Supporting Institution

Conacyt, UNAM, UADY

Project Number

Becas Nacionales 807031, SNI 25997, SNI 16167, SNI SNI 38368; DGAPA-PAPIIT IN101322; FMAT-2023-0002, FMAT-PTA-2022

Thanks

Conacyt, UNAM, UADY

References

• [1] Abe, K., Magid, M.: Relative nullity foliations and indefinite isometric immersions. Pacific J. Math. 142 (1), 1-20 (1986).
• [2] Atindogbe, C., Harouna, M. M., Tossa, J.: Lightlike hypersurfaces in Lorentzian manifolds with constant screen principal curvatures. Afr. Diaspora J. Math. 16 (2), 31-45 (2014).
• [3] Bishop, R., Crittenden, R.: Geometry of manifolds. American Mathematical Society, Providence (2001).
• [4] Bonnet, O.: Mémoire sur la théorie des surfaces applicables. J. Ec. Polyt. 92, 72-92 (1867).
• [5] Atindogbe, C. Ezin, J. P., Tossa, J.: Reduction of the codimension for lightlike isotropic submanifolds. J. Geom. Phys. 42 (1-2), 1-11 (2002).
• [6] Canevari, S., Tojeiro, Ruy.: Isometric immersions of space forms into Sp × R. Math. Nachr. 293 (7), 1259-1277 (2020).
• [7] Chen, Q. and Xiang, C. R.: Isometric immersions into warped product spaces. Acta Math. Sin. (Engl. Ser.) 26 (12), 2269-2282 (2010).
• [8] Dajczer, M.: Submanifolds and isometric immersions. Mathematics Lecture Series. Publish or Perish, Houston (1990).
• [9] Dajczer, M., Onti, C. R., Vlachos, T.: Isometric immersions with flat normal bundle between space forms. Arch. Math. 116 (5), 577-583 (2021).
• [10] Dajczer, M., Tojeiro, R.: Isometric immersions in codimension two of warped products into space forms. Illinois J. Math. 48, (3) 711-746 (2004).
• [11] Dajczer, M., Tojeiro, R.: Submanifold theory: Beyond an introduction. Universitext. Springer, New York (2019).
• [12] Daniel, B.: Isometric immersions into Sn × R and Hn × R and applications to minimal surfaces. Trans. Am. Math. Soc. 361 (12), 6255-6282 (2009).
• [13] Duggal, K. L., Bejancu, A.: Lightlike submanifolds of codimension two. Math. J. Toyama Univ. 15, 59-82 (1992).
• [14] Duggal, K. L., Bejancu, A.: Lightlike submanifolds of semi-Riemannian manifolds and applications. Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1996).
• [15] Duggal, K. L., Sahin, B.: Differential geometry of lightlike submanifolds. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2010).
• [16] Eisenhart, L. P.: Riemannian geometry. Princeton University Press (1964).
• [17] Graves, L. K.: Codimension one isometric immersions between Lorentz spaces. Trans. Am. Math. Soc. 252, 367-392 (1979).
• [18] Greene, R. E.: Isometric embeddings of Riemannian and pseudo-Riemannian manifolds. American Mathematical Society, Providence (1970).
• [19] Gromov, M.: Isometric immersions of Riemannian manifolds. Elie Cartan et les Mathématiques d’Aujourd’hui, Astérisque. 129-133 (1985).
• [20] Jacobowitz, H.: The Gauss-Codazzi equations. Tensor 39, 15-22 (1982).
• [21] Kitamura, S.: The imbedding of spherically symmetric space times in a Riemannian 5-space of constant curvature. Tensor (N.S.) 16, 74-83 (1965).
• [22] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Volume 1. Wiley Interscience (1996).
• [23] Lawn, M. A., Ortega, M.: A fundamental theorem for hypersurfaces in semi-Riemannian warped products. J. Geom. Phys. 90, 55-70 (2015).
• [24] Li, X. X., Zhang, T. Q.: Isometric immersions of higher codimension into the product Sk × Hn+p−k. Acta Math. Sin. (Engl. Ser.) 30 (12), 2146-2160 (2014).
• [25] Lira, J. H., Tojeiro, R. Vitório, F.: A Bonnet theorem for isometric immersions into products of space forms. Arch. Math. 95 (5), 469-479 (2010).
• [26] Magid, M. A.: Isometric immersions of Lorentz space with parallel second fundamental forms. Tsukaba J. Math. 8 (1), 31-54 (1984).
• [27] O’Neill, B.: Semi-Riemannian geometry, with applications to relativity. Academic Press, London (1983).
• [28] Poznjak, E. G., Sokolov, D. D.: Isometric immersions of Riemannian spaces in Euclidean spaces. J. Soviet Math. 14, 1407-1428 (1980).
• [29] Spivak, M.: A comprehensive introduction to differential geometry. Vol. IV. Publish or Perish, Wilmington (1979).
• [30] Tenenblat, K.: On isometric immersions of Riemannian manifolds. Bull. Braz. Math. Soc. 2 (2), 23-36 (1971).
• [31] Tu, L.: Differential geometry: connections, curvature and characteristic classes. Springer-Verlag, New York (2017).