Multiply Warped Product Generalized Semi-Invariant Submanifolds

Year 2021, Volume: 14 Issue: 2, 313 - 330, 29.10.2021

Moctar Traore Hakan Mete Taştan Sibel Gerdan Aydın

https://doi.org/10.36890/iejg.845483

Cited By: 1

Abstract

We define generalized semi-invariant submanifolds in locally product Riemannian manifolds. Then we study multiply warped product generalized semi-invariant submanifolds in the same structure. We give an existence theorem for such submanifolds. We also give necessary and sufficient conditions for such a submanifold to be a multiply direct product submanifold. Moreover, we establish a general inequality for such submanifolds.

Keywords

Multiply warped product submanifolds, slant distribution, invariant distribution, anti-invariant distribution, generalized semi-invariant submanifold, locally product Riemannian manifold

References

• [1] Adati, T. : Submanifolds of an almost product Riemanian manifold. Kodai Math J. 4, 327-343 (1981).
• [2] Al-Solamy, F.R., Khan, M.A.: Warped product submanifolds of Riemannian product manifolds. Hindawi Publishing Corporation Abstract and Applied Analysis. Article ID 724898, 12 pages (2012).
• [3] Atçeken, M.: Warped product semi-slant submanifolds in locally Riemannian product manifolds. Bull. Austral. Math. Soc. 77 (2), 177-186 (2008).
• [4] Atçeken, M.: Warped Product semi-invariant submanifolds in locally decomposable Riemannian Manifolds. Hacet. J. Math. Stat. 40 (3), 401–407 (2011).
• [5] Atçeken, M.: Geometry of warped product semi-invariant submanifolds of a locally Riemannian product manifolds. Serdica Math. J. 35, 273-289 (2009).
• [6] Bejan, C.L.: Almost semi-invariant submanifolds of locally product Riemannian manifolds. Bull. Math. de la Soc. Sci. Math. de la R. S. de Roumanie Tome. 32 (80), No. 1, 3-9 (1988).
• [7] Bejancu, A.: Semi-invariant submanifolds of locally product Riemannian manifolds. An. Univ. Timi¸soara Ser. ¸Stiint. Math. Al. 22(1-2), 3-11 (1984).
• [8] Bishop, R. L., O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145(1), 1-49 (1969).
• [9] Chen, B. Y.: Geometry of warped product submanifolds in Kaehler manifolds. Monatsh Math. 133, 177-195 (2001).
• [10] Chen, B. Y., Dillen, F.: Optimal Inequalities For Multiply Warped Product Submanifolds. Int. Electron. J. Geom. 1(1), 1-11 (2008).
• [11] Chen, B.Y.: Differential geometry of warped product manifolds and submanifolds. World Scientific. (2017).
• [12] Dillen, F., Nölker, S.: Semi-paralellity multi rotation surfaces and the helix property. J. Reine. Angew. Math. 435, 33-63 (1993).
• [13] Gerdan Aydın, S., Taştan, H. M., Traore, M., Ülker, Y.: Biwarped product submanifolds with a slant base factor. (Preprint).
• [14] Liu, X., Shao, F. M.: Skew semi-invariant submanifolds of locally product manifold. Portugalie Math. 56, 319-327 (1999).
• [15] Li, H., Liu, X.: Semi-slant submanifolds of a locally product manifold. Georgian Math. J. 12, 273–282 (2005).
• [16] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press. San Diego (1983).
• [17] Şahin, B.: Slant submanifolds of an almost product Riemannian manifold. J. Korean Math. Soc. 43, 717-732 (2006).
• [18] Şahin, B.: Warped Product semi-slant submanifolds of a locally product Riemannian manifold. Studia Sci. Math. Hungar. 46(2), 169–184 (2009).
• [19] Şahin, B.: Warped product semi-invariant submanifolds of a locally product Riemannian manifold. Bull. Math. Soc. Sci. Math. Roumanie. 49(97), 4, 383-394 (2006).
• [20] Taştan, H. M.: Warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold. Turk. J. Math. 39, 453-466 (2015).
• [21] Taştan, H. M., Özdemir, F.: The geometry of hemi-slant submanifolds of a locally product Riemannian manifold. Turk. J. Math. 39, 268-284 (2015).
• [22] Uddin, S., Mihai, A., Mihai, I., Al-Jedani, A.: Geometry of bi-warped product submanifolds of locally product Riemannian manifolds. RACSAM. 114(42), (2020). https://doi.org/10.1007/s13398-019-00766-6.
• [23] Ünal, B.: Multiply warped products. J. Geom. Phys. 34(3), 287-301 (2000).
• [24] Xu, S., Ni, Y.: Submanifolds of product Riemannian manifolds. Acta Mathematica Scientia. 20B(2), 213-218 (2000).
• [25] Yano, K., Kon, M.: Structures on manifolds. World Scientific, Singapore (1984).