The Examples of Generators of 2-cob Cobordisms in Semi-Riemannian Space R(3,1)

Abstract

In their book published in 1996, Duggal and Bejancu demonstrated the existence of a lightlike (null) subspace in a semi-Riemannian manifold and filled an important gap needed for the geometry of submanifolds. In addition to birational invariants such as uniruledness and kodaira dimensions for Semi-Riemannian manifolds, in this study, by emphasizing the k (U) values which are the numbers of maximum linearly independent lightlike vectors, that are a birational invariance, the examples of generators of 2- Cob cobordisms are given in Semi-Riemannian Space R(3,1). Their codaira dimensions and k (U) invariants of these examples has been expressed.

Keywords

• Cobordism
• Birationel invariant
• Semi-Riemannian manifold
• Kodaira dimension
• Ruled variety

R(3,1) Semi-Riemannian Uzayda 2-Cob Üreteç Kobordizm Örnekleri

Abstract

Duggal ve Bejancu 1996 da yayınladıkları kitapta bir semi-Riemannian manifoldda lightlike (null) alt uzayın varlığını gösterdiler ve alt manifoldların geometrisi için ihtiyaç duyulan önemli bir boşluğu doldurdular. Semi-Riemannian manifoldlar için unireglelik, kodaira boyutu gibi birasyonel invaryantların yanında maximum lineer bağımsız lightlike vektörlerin sayıları olan k(U) değerlerinin de bir birasyonel invaryant olduğu vurgıulanarak R31 Semi-Riemannian Uzayda 2-Cob Üreteç kobordizmlere örnekler verilmiş, bunların kodaira boyutları ve k(U) invaryantları ifade edilmiştir.

Keywords

• Kobordizm
• Birasyonel invaryant
• Semi-Riemannian manifold
• Kodaira boyutu
• Regle yapı

References

1. [1] Guillemin, V., Sternberg, S. Birational equivalence in the symplectic category. Invent. Math. 1989; 97: 485–522.
2. [2] McDuff, D., Salamon, D. Introduction to Symplectic Topology, 2nd edn New York: Oxford Math. Monogr. Oxford University Press;1998.
3. [3] Matsuki,K.:Lectures on factorization of birationalmaps. arXiv:math.AG/0002084
4. [4] Hu J., Li T-J., Ruan Y., Birational cobordism invariance of uniruled symplectic manifolds, Invent. math. 2008; 172: 231–275.
5. [5] Li, T.-J.,Existence of embedded symplectic surfaces. In: Geometry and Topology of Manifolds. Fields Inst. Commun.,47: 203–217. Am. Math. Soc., Providence, RI. 2005.
6. [6] Kollar J., Low degree polynomial equations: arithmetic, geometry, topology. European Congress of Mathematics, vol. I (Budapest, 1996). Prog. Math., 1998; 168: 255–288.
7. [7] Ruan Y.,Virtual neighborhoods and pseudoholomorphic curves. Turk. J. Math., 1999; 23: 161–231.
8. [8] Duggal K. L. and Bejancu, A., Lightlike submanifold of semi-Riemannian manifolds and its applications, The Netherlands: Kluwer Academic; 1996.

9. [9] Duggal K. L. and Şahin, B., Differential geometry of lightlike submanifolds, Birkhäuser: Verlag; 2010.
10. [10] Kupeli D. N., Singular Semi-Riemannian Geometry, Dordrecht:Kluwer Academic Publishers; 1996.
11. [11] Kocaayan H., Karakteristik Sınıfları, [Y.Lisans Tezi], İzmir: Ege Üniversitesi Fen Bilimleri Enstitüsü; 2012.
12. [12] Chern,S.S., Charecteristic classes of Hermitian manifolds, Annals of Mathematics, 1946;47(1):85-121.
13. [13] Çallıalp, F., Soyut Cebir, İstanbul: Birsen Yayınevi; 2009.
14. [14] Pierce, D. [internet], Galois Teorisi, Matematik Bölümü,MSGSÜ, 2018.Available from: http://mat.msgsu.edu.tr/~dpierce/Dersler/Galois-Teorisi/galois-kurami-2018.pdf.
15. [15] O’Neill B., Semi-Riemannian Geometry with Applications to relativity, San Diego-California: Akademic Press. Inc.; 1983.
16. [16] İncesu M., “Benzerlik Geometrisinde Noktaların Tam İnvaryantları Sistemi”, [doktora tezi] , Trabzon: Karadeniz Teknik Üniversitesi, 2008
17. [17] Sağıroğlu Y., Parametrik Eğrilerin Afin Diferensiyel İnvaryantları, [doktora tezi], Trabzon:Karadeniz Teknik Üniv. Fen Bilimleri Ens., 2002.
18. [18] Hirsch M.W., Graduate Text in Mathematics: Differential Topology, New York: Springer; 1976.
19. [19] Stong R. E., Notes on cobordism theory, Princeton N.J.: Princeton University Press; 1968.
20. [20] Thom, R., Qoelques propi_et es globales des varieties differentiables, Commentari Mathematici Helvetici, 1954; 28: 17-86. [21] Ören İ., O(3,1)-Ortogonal Grubu için Noktaların İnvaryantları, [doktora tezi], Trabzon: Karadeniz Teknik Üniversitesi, 2007.
21. [22] Boucksom S., Demailly J.P., Paun M. and Peternell T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Alg. Geometry 2013; 22 (2): 201-248.
22. [23] Bogomolov F., Tschinkel Y., Rational curves and points on K3 surfaces, Amer. J. Math., 2005;127 (4) : 825-835.
23. [24] Shioda T., An example of unirational surfaces in characteristicp . Math. Ann., 1974; 211: 233–236.
24. [25] Wikipedia contributors. Ruled variety [Internet]. Wikipedia, The Free Encyclopedia; 2020 Apr 12, 13:29 UTC [cited 2020 Oct 9]. Available from: https://en.wikipedia.org/w/index.php?title=Ruled_variety&oldid=950515178.