Characterizations of PR-Slant Warped Submanifolds in Para-Kaehler Spaces via Weingarten Maps
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Abstract
This paper investigates $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifolds in para-K\"{a}hler manifolds $\bar{F}^{2m}$. First, fundamental properties of $\mathcal{P}\mathcal{R}$-pseudo-slant submanifolds in $\bar{F}^{2m}$ are obtained. Then non-existence and characterization results for $\mathcal{P}\mathcal{R}$-pseudo-slant warped product submanifolds are derived in terms of the Weingarten and endomorphism mappings, including the corresponding slant factor in $\bar{F}^{2m}$. In addition, a numerical example of the warped product $F_{\lambda}\times_{f}F_{\bot}$ in $\bar{F}^{2m}$ is provided to illustrate the results.
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References
[1] D. (E.) Afanasev and M. (O.) Katanaev, Global properties of warped solutions in general relativity with an electromagnetic field and a cosmological constant, Phys. Rev. D, 100 (2019), 024052.
[2] P. Alegre and A. Carriazo, Slant submanifolds of para-Hermitian manifolds, Mediterr. J. Math., 14 (2017), 214.
[3] S. Ayaz and Y. Gündüzalp, Warped product pointwise hemi-slant submanifolds whose ambient spaces are nearly para-Kaehler manifolds, Turk. J. Math., 48 (2024), 477--497.
[4] S. Ayaz and Y. Gündüzalp, Pointwise bi-slant doubly warped product submanifolds in para-Kaehler manifolds, Filomat, 39 (2025), 4459--4476.
[5] S. Ayaz and Y. Gündüzalp, Geometry of pointwise hemi-slant warped product submanifolds in para-contact manifolds, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 74 (2025), 117--129.
[6] S. Ayaz and Y. Gündüzalp, A note on pointwise semi-slant submanifolds in a para-Kaehler manifold, Int. Electron. J. Geom., 18 (2025), 462--474.
[7] R. (L.) Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1--49.
[8] A. (M.) Blaga, On a sequence of slant submanifolds in almost product Riemannian setting, Int. Electron. J. Geom., 16 (2023), 594--597.
[9] B. (Y.) Chen, Slant immersion, Bull. Austral. Math. Soc., 41 (1990), 135--147.
[10] B. (Y.) Chen and M. (I.) Munteanu, Geometry of $PR$-warped products in para-K"{a}hler manifolds, Taiwanese J. Math., 16 (2012), 1293--1327.
[11] B. (Y.) Chen, S. Shenawy and U. (C.) De, The impact of quasi-conformal curvature tensor on warped product manifolds, Axioms, 13 (2024), 500.
[12] M. Dhiman, A. Kumar and S. (K.) Srivastava, $PR$-semi slant warped product submanifold of para-Kenmotsu manifolds, Results Math., 77 (2022), 142.
[13] Y. Gündüzalp, Warped product pointwise hemi-slant submanifolds of a para-Kaehler manifold, Filomat, 36 (2022), 275--288.
[14] S. Hiepko, Eine innere kennzeichnung der verzerrten produkte, Math. Ann., 241 (1979), 209--215.
[15] S. (T.) Hong, Warped products and black holes, Nuovo Cimento B, 120 (2005), 1227--1234.
[16] A. Kumar and A. Sharma, Warped product pointwise $PR$-semi-slant immersions in a para-K"{a}hler manifold, Bull. Transilv. Univ. Brasov Ser. III Math. Comput. Sci., 5 (2025), 133--146.
[17] P. Libermann, Sur le probl`{e}me d''{e}quivalence de certaines structures infinit'{e}simales, Ann. Mat. Pura Appl., 36 (1954), 27--120.
[18] B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
[19] J. Petean, Indefinite K"{a}hler--Einstein metrics on compact complex surfaces, Commun. Math. Phys., 189 (1997), 227--235.
[20] A. Sharma, Non-existence of $PR$-semi slant warped product submanifold in a para-K"{a}hler manifold, Kyungpook Math. J., 60 (2020), 197--210.
[21] A. Sharma, Pointwise $PR$-pseudo-slant submanifolds of para-Kaehler manifolds, Bull. Transilv. Univ. Brasov Ser. III Math. Comput. Sci., 1 (2021), 231--240.
[22] A. Sharma, A. Kumar, N. Bin Turki and S. Shenawy, Certain investigations of sequential warped product submanifolds on cosymplectic manifolds, J. Inequal. Appl., 2023 (2023), 125.
[23] S. Shenawy, U. (C.) De, N. Bin Turki and N. A. Pundeer, Projective collineations in warped product manifolds and $(PRS)_n$ manifolds, Symmetry, 15 (2023), 1644--1644.
[24] S. (S.) Shukla and P. Kumar, Generalized Cauchy--Riemann screen pseudo-slant lightlike submanifolds of indefinite K"{a}hler manifolds, Asian-Eur. J. Math., 17 (2024), 2450045.
[25] S. (K.) Srivastava and A. Sharma, Pointwise pseudo-slant warped product submanifolds in a Kahler manifold, Mediterr. J. Math., 14 (2017), 20.
[26] S. (K.) Srivastava and A. Sharma, Geometry of $PR$-semi-invariant warped product submanifolds in paracosymplectic manifold, J. Geom., 108 (2017), 61--74.
[27] S. (K.) Srivastava, F. Mofarreh, A. Kumar and A. Ali, Characterizations of $PR$-pseudo-slant warped product submanifold of para-Kenmotsu manifold with slant base, Symmetry, 14 (2022), 1001.