We establish a general inequality for the squared norm of second fundamental form of these types of submanifolds.
📌 3 Key Takeaways
🎯
Objective & Hypothesis: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.
📊
Key Findings & Data: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 $_fM^{\theta}\times_{b}M^{T}$.
💡
Conclusion & Impact:We establish a general inequality for the squared norm of second fundamental form of these types of submanifolds.
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.
Aydın, S. G., Taştan, H. M. (2021). Conformal-twisted product semi-slant submanifolds in globally conformal Kaehler manifolds. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 4. https://doi.org/10.15672/hujms.801651
B.Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds, Monatsh. Math. 133, 177–195, 2001.
B.Y. Chen, Differential geometry of warped product manifolds and submanifolds, World Scientific, 2017.
B.Y. Chen, F. Alghamdi and S. Uddin, Geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds, arXiv:2011.03997, 2020.
S. Dragomir and L. Ornea, Locally conformal Kähler geometry, Progress in Mathematicsi 155, Birkhäuser, Boston Inc: Boston, MA, 1998.
M. Gutierrez and B. Olea, Semi-Riemannian manifolds with a doubly warped structures, Rev. Mat. Iberoam. 28 (1), 1–24, 2012.
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.
K. Matsumoto, Warped product semi-slant submanifolds in locally conformal Kaehler manifolds, Proc. Int. Geom. Cent. 10 (2), 8–23, 2017.
K. Matsumoto, Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II, Proc. Int. Geom. Cent. 11 (3), 27–44, 2018.
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.
N. Papaghiuc, Semi-slant submanifolds of a Kählerian manifold, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 40, 55–61, 1994.
R. Ponge and H. Reckziegel, Twisted products pseudo- Riemannian geometry, Geom. Dedicata, 48, 15–25, 1993.
B. Şahin, Nonexistence of warped product semi-slant submanifolds of Kaehler manifolds, Geom. Dedicata, 117, 195–202, 2005.
B. Şahin, Notes on doubly warped and doubly twisted product CR-submanifolds of Kaehler manifolds, Mat. Vesnik, 59, 205–210, 2007.
B. Şahin, Warped product semi-slant submanifolds of a locally product Riemannian manifold, Studia Sci. Math. Hungar. 46 (2), 169–184, 2009.
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.
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.
S. Uddin, On doubly warped and doubly twisted product submanifolds, Int. Electron. J. Geom. 3 (1), 35–39, 2010.