Conformal-twisted product semi-slant submanifolds in globally conformal Kaehler manifolds

Year 2021, , 1028 - 1046, 06.08.2021

Sibel Gerdan Aydın , Hakan Mete Taştan

https://doi.org/10.15672/hujms.801651

Abstract

We introduce the notion of conformal-twisted product submanifolds of the form $_fM^{T}\times_{b}M^{\theta}$ and $_fM^{\theta}\times_{b}M^{T}$, where $M^T$ is a holomorphic submanifold and $M^\theta$ is a proper slant submanifold of $M$ in a globally conformal Kaehler manifold and $f$ and $b$ are conformal factor and twisting function, respectively. We give necessary and sufficient conditions for proper semi-slant submanifold to be a locally conformal-twisted product for such submanifolds of the form $_fM^{T}\times_{b}M^{\theta}$ and $_fM^{\theta}\times_{b}M^{T}$. We establish a general inequality for the squared norm of second fundamental form of these types of submanifolds.

Keywords

twisted product manifold, holomorphic ditribution, locally and globally conformal Kaehler manifold, semi-slant submanifold, slant distribution

Supporting Institution

TUBITAK

Project Number

119F179

References

• [1] A. Bejancu, CR-Submanifolds of Kaehler manifold, Proc. Amer. Math. Soc. 69, 135- 142, 1978.
• [2] A. Bejancu , Semi-invariant submanifolds of locally product Riemannian manifolds, An. Univ. West. Timis. Ser. Mat.-Inform. 22, 1984.
• [3] R.L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 1 (145), 1–49, 1969.
• [4] V. Bonanzinga and K. Matsumoto, Twisted product CR-submanifolds in a locally conformal Kaehler space forms, Filomat, 34 (6), 1915–1925, 2020.
• [5] B.Y. Chen, Geometry of submanifolds and its applications, Science University of Tokyo, 1981.
• [6] B.Y. Chen, Geometry of slant submanifolds, Katholieke Universiteit, Leuven, Leuven 1990.
• [7] B.Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds, Monatsh. Math. 133, 177–195, 2001.
• [8] B.Y. Chen, Differential geometry of warped product manifolds and submanifolds, World Scientific, 2017.
• [9] B.Y. Chen, F. Alghamdi and S. Uddin, Geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds, arXiv:2011.03997, 2020.
• [10] S. Dragomir and L. Ornea, Locally conformal Kähler geometry, Progress in Mathematicsi 155, Birkhäuser, Boston Inc: Boston, MA, 1998.
• [11] M. Gutierrez and B. Olea, Semi-Riemannian manifolds with a doubly warped structures, Rev. Mat. Iberoam. 28 (1), 1–24, 2012.
• [12] M.A. Lone, A lower bound of normalized scalar curvature for the submanifolds of locally conformal Kaehler space form using Casorati curvatures, Filomat, 31 (15), 4925–4932, 2017.
• [13] K. Matsumoto, Warped product semi-slant submanifolds in locally conformal Kaehler manifolds, Proc. Int. Geom. Cent. 10 (2), 8–23, 2017.
• [14] K. Matsumoto, Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II, Proc. Int. Geom. Cent. 11 (3), 27–44, 2018.
• [15] M.I. Munteanu, Doubly warped product CR-submanifolds in locally conformal Kähler manifolds, Monatsh. Math. 150, 333–342, 2007.
• [16] M.I. Munteanu, A note on doubly warped product contact CR-submanifolds in trans- Sasakian manifolds, Acta Math. Hungar. 116 (1-2), 121–126, 2007.
• [17] N. Papaghiuc, Semi-slant submanifolds of a Kählerian manifold, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 40, 55–61, 1994.
• [18] R. Ponge and H. Reckziegel, Twisted products pseudo- Riemannian geometry, Geom. Dedicata, 48, 15–25, 1993.
• [19] B. Şahin, Nonexistence of warped product semi-slant submanifolds of Kaehler manifolds, Geom. Dedicata, 117, 195–202, 2005.
• [20] B. Şahin, Notes on doubly warped and doubly twisted product CR-submanifolds of Kaehler manifolds, Mat. Vesnik, 59, 205–210, 2007.
• [21] B. Şahin, Warped product semi-slant submanifolds of a locally product Riemannian manifold, Studia Sci. Math. Hungar. 46 (2), 169–184, 2009.
• [22] H.M. Taştan and S. Gerdan, Doubly twisted product semi-invariant submanifolds of a locally product Riemannian manifold, Mathematical Advances in Pure and Applied Sciences, 1 (1), 23–26, 2018.
• [23] H.M. Taştan and M.M. Tripathi, Semi-slant submanifolds of a locally conformal Kaehler manifold, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) Mathematics Tomul LXII, 1 (2), 337–347, 2016.
• [24] S. Uddin, On doubly warped and doubly twisted product submanifolds, Int. Electron. J. Geom. 3 (1), 35–39, 2010.
• [25] B. Ünal, Doubly warped products, Differential Geom. Appl. 15, 253–263, 2001.
• [26] I. Vaisman, On locally and globally conformal Kähler manifolds, Trans. Amer. Math. Soc. 262 (2), 533–542, 1980.
• [27] G.E. Vilcu, Ruled CR-submanifolds of locally conformal Kähler manifolds, J. Geom. Phys. 62 (6), 1366–1372, 2012.
• [28] G.E. Vilcu, Horizontally conformal submersions from CR-submanifolds of locally conformal Kaehler manifolds, Mediterr. J. Math. 17 (26), 2020.
• [29] K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore, 1984.