The Stability Problem of Certain Anti-Invariant Submanifolds in Golden Riemannian Manifolds

Yıl 2023, Cilt: 16 Sayı: 1, 343 - 348, 30.04.2023

Mustafa Gök , Erol Kılıç

https://doi.org/10.36890/iejg.1240437

Cited By: 1

Öz

In this study, we discuss the stability of some anti-invariant submanifolds of golden Riemannian manifolds under certain conditions in terms of the Ricci curvature tensors of the ambient manifold and the submanifold.

Anahtar Kelimeler

Stability, metallic Riemannian manifold, anti-invariant submanifold

Kaynakça

• [1] Bahadır, O., Uddin, S.: Slant submanifolds of golden Riemannian manifolds. J. Math. Ext. 13(4), 23-39 (2019).
• [2] Bejancu, A.: Geometry of CR Submanifolds. D. Reidel Publishing Company. Dordrecht (1986).
• [3] Blaga, A.M.: The geometry of golden conjugate connections. Sarajevo J. Math. 10(23), 237-245 (2014).
• [4] Blaga, A.M., Hreţcanu, C.E.: Golden warped product Riemannian manifolds. Libertas Math. 37(1), 1-11 (2017).
• [5] Blaga, A.M., Hreţcanu, C.E.: Invariant, anti-invariant and slant submanifolds of a metallic Riemannian manifold. Novi Sad J. Math. 48(2), 55-80 (2018).
• [6] Cîrnu, M.: Stability of slant and semi-slant submanifolds in Sasaki manifolds. Acta Univ. Apulensis 20, 63-78 (2009).
• [7] Chen, B.Y., Morvan, J.M.: Deformations of isotropic submanifolds in Kahler manifolds. J. Geom. Phys. 13(1), 79-104 (1994).
• [8] Chen, B.Y., Leung, P.F., Nagano, T.: Totally geodesic submanifolds of symmetric spaces, III. Preprint, (1980).
• [9] Crâşmăreanu, M.C., Bercu, G.: Semi-invariant submanifolds in metric geometry of endomorphisms. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90(1), 87–92 (2020).
• [10] Crâşmăreanu, M.C., Hreţcanu, C.E.: Golden differential geometry. Chaos Solitons Fractals 38(5), 1229-1238 (2008).
• [11] Crâşmăreanu, M.C., Hreţcanu, C.E., Munteanu, M.I.: Golden- and product-shaped hypersurfaces in real space forms. Int. J. Geom. Methods Mod. Phys. 10(4), Article ID 1320006, 9 pages (2013).
• [12] Erdoğan, F.E., Yıldırım, C.: On a study of the totally umbilical semi-invariant submanifolds of golden Riemannian manifolds. J. Polytechnic 21(4), 967-970 (2018).
• [13] Goldberg, S.I., Yano, K.: Polynomial structures on manifolds. Kodai Math. Sem. Rep. 22(2), 199-218 (1970).
• [14] Gök, M., Keleş S., Kılıç, E.: Some characterizations of semi-invariant submanifolds of golden Riemannian manifolds. Mathematics 7(12), Article ID 1209, 12 pages (2019).
• [15] Gök, M., Keleş, S., Kılıç, E.: Invariant submanifolds in golden Riemannian manifolds. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69(2), 125-138 (2020).
• [16] Gök, M., Kılıç, E., Keleş, S.: Anti-invariant submanifolds of locally decomposable golden Riemannian manifolds. Balk. J. Geom. Appl. 25(1), 47-60 (2020).
• [17] Hreţcanu, C.E., Blaga, A.M.: Submanifolds in metallic Riemannian manifolds. Differ. Geom. Dyn. Syst. 20, 83-97 (2018).
• [18] Hreţcanu, C.E., Blaga, A.M.: Slant and semi-slant submanifolds in metallic Riemannian manifolds. J. Funct. Spaces 2018, Article ID 2864263, 13 pages (2018).
• [19] Hreţcanu, C.E., Blaga, A.M.: Hemi-slant submanifolds in metallic Riemannian manifolds. Carpathian J. Math. 35(1), 59-68 (2019).
• [20] Hreţcanu, C.E., Crâşmăreanu, M.C.: On some invariant submanifolds in a Riemannian manifold with golden structure. An. Ştiinţ. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 53(suppl. 1), 199-211 (2007).
• [21] Hreţcanu, C.E., Crâşmăreanu, M.C.: Applications of the golden ratio on Riemannian manifolds. Turk. J. Math. 33(2), 179-191 (2009).
• [22] Hreţcanu, C.E., Crâşmăreanu, M.C.: Metallic structures on Riemannian manifolds. Rev. Un. Mat. Argentina 54(2), 15-27 (2013).
• [23] Lawson, H.B.: Minimal Varieties in Real and Complex Geometry. Presses de l’Université de Montréal, Montreal (1974).
• [24] Oh, Y.G.: Second variation and stability of minimal Lagrangian submanifolds. Invent. Math. 101(2), 501-519 (1990).
• [25] Özgür, C., Özgür, N.Y.: Classification of metallic shaped hypersurfaces in real space forms. Turk. J. Math. 39(5), 784-794 (2015).
• [26] Özgür, C., Özgür, N.Y.: Metallic shaped hypersurfaces in Lorentzian space forms. Rev. Un. Mat. Argentina 58(2), 215-226 (2017).
• [27] Özkan, M.: Prolongations of golden structures to tangent bundles. Differ. Geom. Dyn. Syst. 16, 227-238 (2014).
• [28] Pitiş, G.: On some submanifolds of a locally product manifold. Kodai Math. J. 9(3), 327-333 (1986).
• [29] Renteln, P.: Manifolds, Tensors, and Forms. Cambridge University Press. New York (2014).
• [30] Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker. Newyork (1970).
• [31] Yano, K., Kon, M.: Structures on Manifolds. World Scientific. Singapore (1984).