Construction of Networks by Associating with Submanifolds of Almost Hermitian Manifolds

Abstract

The idea that data lies in a non-linear space has brought up the concept of manifold learning as a part of machine learning and such notion is one of the most important research fields of today. The main idea here is to design the data as a submanifold model embedded in a high-dimensional manifold. On the other hand, graph theory is one of the most important research areas of applied mathematics and computer science. As a result, many researchers investigate new methods for machine learning on graphs. From the above information, it is seen that the theory of submanifolds and graph theory have become two important concepts in machine learning and nowadays, the geometric deep learning research area using these two concepts has emerged. By combining these two fields, this article aims to present the relationships between submanifolds of complex manifolds with the help of graphs. In this paper, we build some directed networks by identifying with submanifolds of almost Hermitian manifolds. Moreover, we give some results and relations among holomorphic submanifolds, totally real submanifolds, CR-submanifolds, slant submanifolds, semi-slant submanifolds, hemi-slant submanifolds, and bi-slant submanifolds in almost Hermitian manifolds in terms of graph theory.

Keywords

• Almost Hermitian manifold
• Digraphs
• Graph theory
• Machine learning
• Manifold learning
• Submanifolds

References

1. [1] M. Aquib, J. W. Lee, G. E. Vilcu, D. W. Yoon, Classification of Casorati ideal Lagrangian submanifolds in complex space forms, Differ. Geom. Appl., 63 (2019), 30-49.
2. [2] J. Lee, G. E. Vˆılcu, Inequalities for generalized normalized d-Casorati curvatures of slant submanifolds in quaternionic space forms, Taiwanese J. Math., 19(3) (2015), 691-702.
3. [3] W. M. Othman, S. A. Qasem, C. Ozel, Characterizations of contact CR-warped products of nearly cosymplectic manifolds in terms of endomorphisms, Int. J. Maps Math., 1(2) (2018), 137-153.
4. [4] G. E. Vˆılcu, An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature, J. Math. Anal., 465(2) (2018), 1209-1222.
5. [5] S. K. Yadav, S. K. Chaubey, On Hermitian manifold satisfying certain curvature conditions, Int. J. Maps Math., 3(1) (2020), 10-27.
6. [6] F. Etayo, On quasi-slant submanifolds of an almost Hermitian manifold, Publ. Math. Debrecen, 53(1-2) (1998), 217-223.
7. [7] B. Y. Chen, O. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turk. J. Math., 36(4) (2012), 630-640.
8. [8] C. Murathan, B. Sahin, A study of Wintgen like inequality for submanifolds in statistical warped product manifolds, J. Geom., 109(2) (2018), 1-18.

9. [9] N. Zheng, J. Xue, Statistical Learning and Pattern Analysis for Image and Video Processing, Springer Science & Business Media, 2009.
10. [10] R. A. Fiorini, Computerized tomography noise reduction by CICT optimized exponential cyclic sequences (OECS) co-domain, Fundam. Inform. 141(2–3) (2015), 115-34.
11. [11] F. Monti, D. Boscaini, J. Masci, E. Rodola, J. Svoboda, M. M. Bronstein, Geometric deep learning on graphs and manifolds using mixture model cnns, In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2017) (2017), (pp. 5115-5124).
12. [12] M. Shahzad, F. Sultan, S. I. A. Shah, M. Ali, H. A. Khan, W. A. Khan, Physical assessments on chemically reacting species and reduction schemes for the approximation of invariant manifolds, J. Mol. Liq., 285 (2019), 237-243.
13. [13] M. Shahzad, F. Sultan, M. Ali, W. A. Khan, S. Mustafa, Modeling multi-route reaction mechanism for surfaces: a mathematical and computational approach, Appl. Nanosci., 10(12) (2020), 5069-5076.
14. [14] A. Carriazo, L. Fernandez, Submanifolds associated with graphs, Proc. Am. Math. Soc., 132(11) (2004), 3327-3336.
15. [15] L. Boza, A. Carriazo, L. M. Fernandez, Graphs associated with vector spaces of even dimension: A link with differential geometry, Linear Algebra Appl., 437(1) (2012), 60-76.
16. [16] A. Carriazo, L. M. Fern´andez, A. Rodr´ıguez Hidalgo, Submanifolds weakly associated with graphs, Proc. Indian Acad. Sci.: Math. Sci., 119(3) (2009), 297-318.
17. [17] M. Aquib, Some inequalities for submanifolds in Bochner-Kaehler manifold, Balk. J. Geom. Appl., 23(1) (2018), 1-13.
18. [18] M. Aquib, M. N. Boyom, M. H. Shahid, G. E. Vilcu, The first fundamental equation and generalized Wintgen-type inequalities for submanifolds in generalized space forms, Mathematics, 7 (2019), 1151.
19. [19] J. Bang-Jensen, G. Z. Gutin,Digraphs: Theory, Algorithms and Applications, Springer Science & Business Media, 2008.
20. [20] R. Belmonte, T. Hanaka, I. Katsikarelis, E. J. Kim, M. Lampis, New results on directed edge dominating set, arXiv preprint, (2019), arXiv:1902.04919.
21. [21] J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Macmillan London, 1976.
22. [22] G. Chartrand, L. Lesniak, P. Zhang, Graphs & Digraphs, Chapman and Hall/CRC, 2010.
23. [23] T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, MIT press, 2009.
24. [24] K. H. Rosen, K. Krithivasan, Discrete Mathematics and its Applications, McGraw-Hill, 2013.
25. [25] G. E. Vˆılcu, B. Y. Chen, Inequalities for slant submanifolds in quaternionic space forms, Turk. J. Math., 34 (2010), 115-128.
26. [26] R. Sedgewick, K. Wayne, Algorithms, Addison-Wesley, 4th Edition, 2015.
27. [27] G. Chartrand, S. Tian, Distance in digraphs, Computers & Mathematics with Applications, 34(11) (1997), 15-23.
28. [28] C. W. Lee, Domination in digraphs, J. Korean Math. Soc., 35(4) (1998), 843-853.
29. [29] C. Pang, R. Zhang, Q. Zhang, J. Wang, Dominating sets in directed graphs, Inf. Sci., 180(19) (2010), 3647-3652.
30. [30] R. C. Vandell, Integrity of Digraphs, Dissertations, 1996.
31. [31] K. Yano, M. Kon, Structures on Manifolds, World Scientific, 1985.
32. [32] A. Bejancu, Geometry of CR-submanifolds, Springer Science & Business Media, 2012.
33. [33] B. Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, 1990.
34. [34] A. Carriazo, Bi-slant immersions, In: Proceedings ICRAMS 2000, Kharagpur, India, 2000, 88-97.
35. [35] B. Hassanzadeh, Semi-symmetric metric connection on cosymplectic manifolds, Int. J. Maps Math., 3(2) (2020), 100-108.
36. [36] A. Sa˘glamer, N. C¸ alıs¸kan, On semi-invariant submanifolds of trans-Sasakian Finsler manifolds, Fundamental J. Math. Appl., 1(2) (2018), 112-117.
37. [37] B. Sahin, Warped product submanifolds of Kaehler manifolds with a slant factor, Ann. Pol. Math., 95 (2009), 207-226.
38. [38] N. Papaghiuc, Semi-slant submanifolds of a Kaehlerian manifold, An. Stiint. Univ. Al. I. Cuza Iasi Secct. I a Mat., 40 (1994), 55-61.