In this paper we introduce the concept of pointwise bi-slant submanifolds of locally product Riemannian manifolds and studied warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds. We obtain some characterization results for warped products pointwise bi-slant submanifolds. Also, we provide some non-trivial examples of such warped product submanifolds.
[1] F. R. Al-Solamy, V. A. Khan and S. Uddin, Geometry of warped product semi-slant
submanifolds of nearly Kaehler manifolds, Results in Mathematics. 71(3), 783-799,
2017.
[2] M. Atceken, Warped product semi-slant submanifolds in locally Riemannian product
manifolds, Bull. Aust. Math. Soc. 77(2), 177-186, 2008.
[3] M. Atçeken, Slant submanifolds of a Riemannian product manifold, Acta Math. Sci.
Ser B Engl. Ed. 30(1), 215-224, 2010.
[4] A. Bejancu, Semi-invariant submanifolds of locally product Riemannian manifolds,
An. Univ. Timisoara. Ser Stiint. Math. 22, 3-11, 1984.
[5] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Am. Math. Soc.
145, 1-49, 1969.
[6] J. L. Cabrerizo, A. Carriazo, L.M. Fernandez and M. Fernandez, Semi-Slant submanifolds
of a Sasakian manifold, Geom. Dedicata. 78(2), 183-199, 1999.
[7] B. Y. Chen, Slant immersions, Bull. Aust, Math. Soc. 41(1), 135-147, 1990.
[8] B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds,
Monatsh. Math. 133, 177-195, 2001.
[9] B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds II,
Monatsh. Math. 134, 103-119, 2001.
[10] B. Y. Chen, Another general inequality for CR-warped products in complex space
forms, Hokkaido Math. J. 32(2), 415-444, 2003.
[11] B. Y. Chen, CR-warped products in complex projective spaces with compact holomorphic
factor, Monatshefte für Mathematik. 141(3), 177-186, 2004.
[12] B. Y. Chen, Geometry of warped product submanifolds: A survey, J. Adv. Math. Stud.
6(2), 1-8, 2013.
[13] B. Y. Chen, Differential geometry of warped product manifolds and submanifolds,
World Scientific. 2017.
[14] B. Y. Chen and O. J. Garay, Pointwise slant submanifolds in almost Hermitian manifolds,
Turk. J. Math. 36, 630-640, 2012.
[15] B. Y. Chen, M. H. Shahid and F. R. Al-Solamy, Complex Geometry of Slant Submanifolds,
Springer. Berlin 2021.
[16] B. Y. Chen and S. Uddin, Warped product pointwise bi-slant submanifolds of Kaehler
manifolds, Publicationes Mathematicae. 92(1-2), 183-199, 2018.
[17] F. Etayo, On quasi-slant submanifolds of an almost Hermitian manifold, Publ. Math.
Debrecen. 53(1-2), 217-223, 1998.
[18] M. Gülbahar, E. Kilic and S. S. ÇELK Special proper pointwise slant surfaces of a
locally product Riemannian manifold, Turk. J. Math. 39(6), 884-899, 2015.
[19] S. Hiepko, Eine inner kennzeichungder verzerrten produkte, Math. Ann. 241, 209-215,
1979.
[20] Y. Kentaro and M. Kon, Generic submanifolds, Annali di Matematica Pura ed Applicata.
123(1), 59-92, 1980.
[21] V. A. Khan and M. A. Khan, Semi-slant submanifolds of a nearly Kaehler manifold,
Turk. J. Math. 31(4), 341-353, 2007.
[22] H. Li and X. Liu, Semi-slant submanifolds of a locally product manifold, Georgian
Math. J. 12, 273-282, 2005.
[23] M. A. Lone, M. H. Shahid and S. D. Sharma, Hemi-slant submanifolds of nearly
Kaehler manifolds, Gulf Journal of Mathematics, 4(3), 2016.
[24] M. F. Naghi, M. S. Stankovic and F. Alghamdi, Chens improved inequality for pointwise
hemi-slant warped products in Kaehler manifolds, Filomat. 34(3), 807-814, 2020.
[25] N. Papaghuic, Semi-slant submanifolds of a Kahlerian manifold, An. Stiint. Univ. Al.
I. Cuza Iasi Mat. 40(1), 55-61, 1994.
[26] B. Sahin, Slant submanifolds of an almost product Riemannian manifold, J. Korean
Math. Soc. 43(4), 717-732, 2006.
[27] B. Sahin, Non-existence of warped product semi-slant submanifolds of Kaehler manifolds,
Geometriae Dedicata. 117(1), 195-202, 2006.
[28] B. Sahin, Warped product semi-slant submanifolds of a locally product Riemannian
manifold, Studia. Sci. Math. Hungar. 46(2), 169-184, 2009.
[29] B. Sahin, Warped product pointwise semi-slant submanifolds of Kaehler manifolds,
Port. Math. 70, 252-268, 2013.
[30] B. Sahin and M. Atceken, Semi-invariant submanifolds of Riemannian product manifold,
Balkan J. Geom. Appl. 8(1), 91-100, 2003.
[31] S. Uddin, F. Alghamdi and F. R. Al-Solamy, Geometry of warped product pointwise
semi-slant submanifolds of locally product Riemannian manifolds, Journal of Geometry
and Physics. 152, 103658, 2020.
[32] S. Uddin, B. Y. Chen and F. R. Al-Solamy, Warped product bi-slant immersions in
Kaehler manifolds, Mediterr. J. Math. 14(2), 14-95, 2017.
[33] K. Yano and M. Kon, Structures on manifolds, Series in Pure mathematics, World
scientific Publishing Co. Singapore, 1984.
[1] F. R. Al-Solamy, V. A. Khan and S. Uddin, Geometry of warped product semi-slant
submanifolds of nearly Kaehler manifolds, Results in Mathematics. 71(3), 783-799,
2017.
[2] M. Atceken, Warped product semi-slant submanifolds in locally Riemannian product
manifolds, Bull. Aust. Math. Soc. 77(2), 177-186, 2008.
[3] M. Atçeken, Slant submanifolds of a Riemannian product manifold, Acta Math. Sci.
Ser B Engl. Ed. 30(1), 215-224, 2010.
[4] A. Bejancu, Semi-invariant submanifolds of locally product Riemannian manifolds,
An. Univ. Timisoara. Ser Stiint. Math. 22, 3-11, 1984.
[5] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Am. Math. Soc.
145, 1-49, 1969.
[6] J. L. Cabrerizo, A. Carriazo, L.M. Fernandez and M. Fernandez, Semi-Slant submanifolds
of a Sasakian manifold, Geom. Dedicata. 78(2), 183-199, 1999.
[7] B. Y. Chen, Slant immersions, Bull. Aust, Math. Soc. 41(1), 135-147, 1990.
[8] B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds,
Monatsh. Math. 133, 177-195, 2001.
[9] B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds II,
Monatsh. Math. 134, 103-119, 2001.
[10] B. Y. Chen, Another general inequality for CR-warped products in complex space
forms, Hokkaido Math. J. 32(2), 415-444, 2003.
[11] B. Y. Chen, CR-warped products in complex projective spaces with compact holomorphic
factor, Monatshefte für Mathematik. 141(3), 177-186, 2004.
[12] B. Y. Chen, Geometry of warped product submanifolds: A survey, J. Adv. Math. Stud.
6(2), 1-8, 2013.
[13] B. Y. Chen, Differential geometry of warped product manifolds and submanifolds,
World Scientific. 2017.
[14] B. Y. Chen and O. J. Garay, Pointwise slant submanifolds in almost Hermitian manifolds,
Turk. J. Math. 36, 630-640, 2012.
[15] B. Y. Chen, M. H. Shahid and F. R. Al-Solamy, Complex Geometry of Slant Submanifolds,
Springer. Berlin 2021.
[16] B. Y. Chen and S. Uddin, Warped product pointwise bi-slant submanifolds of Kaehler
manifolds, Publicationes Mathematicae. 92(1-2), 183-199, 2018.
[17] F. Etayo, On quasi-slant submanifolds of an almost Hermitian manifold, Publ. Math.
Debrecen. 53(1-2), 217-223, 1998.
[18] M. Gülbahar, E. Kilic and S. S. ÇELK Special proper pointwise slant surfaces of a
locally product Riemannian manifold, Turk. J. Math. 39(6), 884-899, 2015.
[19] S. Hiepko, Eine inner kennzeichungder verzerrten produkte, Math. Ann. 241, 209-215,
1979.
[20] Y. Kentaro and M. Kon, Generic submanifolds, Annali di Matematica Pura ed Applicata.
123(1), 59-92, 1980.
[21] V. A. Khan and M. A. Khan, Semi-slant submanifolds of a nearly Kaehler manifold,
Turk. J. Math. 31(4), 341-353, 2007.
[22] H. Li and X. Liu, Semi-slant submanifolds of a locally product manifold, Georgian
Math. J. 12, 273-282, 2005.
[23] M. A. Lone, M. H. Shahid and S. D. Sharma, Hemi-slant submanifolds of nearly
Kaehler manifolds, Gulf Journal of Mathematics, 4(3), 2016.
[24] M. F. Naghi, M. S. Stankovic and F. Alghamdi, Chens improved inequality for pointwise
hemi-slant warped products in Kaehler manifolds, Filomat. 34(3), 807-814, 2020.
[25] N. Papaghuic, Semi-slant submanifolds of a Kahlerian manifold, An. Stiint. Univ. Al.
I. Cuza Iasi Mat. 40(1), 55-61, 1994.
[26] B. Sahin, Slant submanifolds of an almost product Riemannian manifold, J. Korean
Math. Soc. 43(4), 717-732, 2006.
[27] B. Sahin, Non-existence of warped product semi-slant submanifolds of Kaehler manifolds,
Geometriae Dedicata. 117(1), 195-202, 2006.
[28] B. Sahin, Warped product semi-slant submanifolds of a locally product Riemannian
manifold, Studia. Sci. Math. Hungar. 46(2), 169-184, 2009.
[29] B. Sahin, Warped product pointwise semi-slant submanifolds of Kaehler manifolds,
Port. Math. 70, 252-268, 2013.
[30] B. Sahin and M. Atceken, Semi-invariant submanifolds of Riemannian product manifold,
Balkan J. Geom. Appl. 8(1), 91-100, 2003.
[31] S. Uddin, F. Alghamdi and F. R. Al-Solamy, Geometry of warped product pointwise
semi-slant submanifolds of locally product Riemannian manifolds, Journal of Geometry
and Physics. 152, 103658, 2020.
[32] S. Uddin, B. Y. Chen and F. R. Al-Solamy, Warped product bi-slant immersions in
Kaehler manifolds, Mediterr. J. Math. 14(2), 14-95, 2017.
[33] K. Yano and M. Kon, Structures on manifolds, Series in Pure mathematics, World
scientific Publishing Co. Singapore, 1984.
Majeed, P., & Lone, M. (2025). Warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds. Hacettepe Journal of Mathematics and Statistics, 54(1), 200-212. https://doi.org/10.15672/hujms.1211053
AMA
Majeed P, Lone M. Warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds. Hacettepe Journal of Mathematics and Statistics. February 2025;54(1):200-212. doi:10.15672/hujms.1211053
Chicago
Majeed, Prınce, and Mehraj Lone. “Warped Product Pointwise Bi-Slant Submanifolds of Locally Product Riemannian Manifolds”. Hacettepe Journal of Mathematics and Statistics 54, no. 1 (February 2025): 200-212. https://doi.org/10.15672/hujms.1211053.
EndNote
Majeed P, Lone M (February 1, 2025) Warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds. Hacettepe Journal of Mathematics and Statistics 54 1 200–212.
IEEE
P. Majeed and M. Lone, “Warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 1, pp. 200–212, 2025, doi: 10.15672/hujms.1211053.
ISNAD
Majeed, Prınce - Lone, Mehraj. “Warped Product Pointwise Bi-Slant Submanifolds of Locally Product Riemannian Manifolds”. Hacettepe Journal of Mathematics and Statistics 54/1 (February 2025), 200-212. https://doi.org/10.15672/hujms.1211053.
JAMA
Majeed P, Lone M. Warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds. Hacettepe Journal of Mathematics and Statistics. 2025;54:200–212.
MLA
Majeed, Prınce and Mehraj Lone. “Warped Product Pointwise Bi-Slant Submanifolds of Locally Product Riemannian Manifolds”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 1, 2025, pp. 200-12, doi:10.15672/hujms.1211053.
Vancouver
Majeed P, Lone M. Warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds. Hacettepe Journal of Mathematics and Statistics. 2025;54(1):200-12.