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Slant Lightlike Submanifolds of Indefinite Kaehler Manifolds and Their Warped Product Manifolds

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Complex Geometry of Slant Submanifolds

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Abstract

As a generalization of both holomorphic and totally real immersions, slant immersions of a Riemannian manifold into an almost Hermitian manifold were introduced by Chen in [8, 9]. The study of slant immersions into complex Euclidean spaces \(\mathbb {C}^{2}\) and \(\mathbb {C}^{4}\) was presented by Chen and Tazawa in [7, 10], while slant immersions of Kaehler \(\mathbb {C}\)-spaces into complex projective spaces were given by Maeda et al. in [29]. Slant immersions of a Riemannian manifold into an almost contact metric manifold were introduced by Lotta in [27]. Then the geometry of slant immersions and slant submanifolds attained momentum after the research on this subject matter by [2, 30, 38] and many other references therein.

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References

  1. Arnold, V.I.: Contact geometry: the geometrical method of Gibbs’s Thermodynamics. In: Proceedings of the Gibbs Symposium, pp. 163–179. Yale University (1989)

    Google Scholar 

  2. Arslan, K., Carriazo, A., Chen, B.Y., Murathan, C.: On slant submanifolds of neutral Kaehler manifolds. Taiwanese J. Math. 14, 561–584 (2010)

    Article  MathSciNet  Google Scholar 

  3. Barros, M., Romero, A.: Indefinite Kaehler manifolds. Math. Ann. 261, 55–62 (1982)

    Article  MathSciNet  Google Scholar 

  4. Bejan, C.L., Duggal, K.L.: Global lightlike manifolds and harmonicity. Kodai Math. J. 28, 131–145 (2005)

    Article  MathSciNet  Google Scholar 

  5. Bishop, R.L., O’Neill, B: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    Google Scholar 

  6. Cartan, E.: Lecons Sur la Ge’ome’trie des Espaces de Riemann. Gauthiers, Paris (1946)

    Google Scholar 

  7. Chen, B.Y., Tazawa, Y.: Slant submanifolds in complex Euclidean spaces. Tokyo J. Math. 14, 101–120 (1991)

    Article  MathSciNet  Google Scholar 

  8. Chen, B.Y.: Slant immersions. Bull. Austral. Math. Soc. 41, 135–147 (1990)

    Article  MathSciNet  Google Scholar 

  9. Chen, B.Y.: Geometry of Slant Submanifolds. Katholieke Universiteit, Leuven (1990)

    Google Scholar 

  10. Chen, B.Y., Tazawa, Y.: Slant surfaces with codimension 2. Ann. Fac. Sci. Toulouse Math. 11, 29–43 (1990)

    Article  MathSciNet  Google Scholar 

  11. Chen, B.Y.: Geometry of warped product \(CR\) -submanifolds in Kaehler manifolds I. Monatsh. Math. 133, 177–195 (2001)

    Article  MathSciNet  Google Scholar 

  12. Chen, B.Y.: Geometry of warped product \(CR\)-submanifolds in Kaehler manifolds II. Monatsh. Math. 134, 103–119 (2001)

    Article  MathSciNet  Google Scholar 

  13. Chen, B.Y., Garay, O.J.: Pointwise slant submanifolds in almost Hermitian manifolds. Turkish. J. Math. 35, 1–11 (2011)

    MathSciNet  Google Scholar 

  14. Duggal, K.L., Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Kluwer Academic Publishers (1996)

    Google Scholar 

  15. Duggal, K.L., Jin, D.H.: Totally umbilical lightlike submanifolds. Kodai Math. J. 26, 49–68 (2003)

    Article  MathSciNet  Google Scholar 

  16. Duggal, K.L., Sahin, B.: Screen Cauchy Riemann lightlike submanifolds. Acta Math. Hungar. 106(1–2), 137–165 (2005)

    Article  MathSciNet  Google Scholar 

  17. Duggal, K.L., Sahin, B.: Differential Geometry of Lightlike Submanifolds. Birkhauser Verlag AG, Berlin (2010)

    Google Scholar 

  18. Etayo, F.: On quasi-slant submanifolds of an almost Hermitian manifold. Publ. Math. Debrecen. 53, 217–223 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Fritelli, S., Kamran, N., Kozamer, C., Newman, E.T.: Null surfaces and contact geometry. J. Hyperbolic Differ. Equ. 2(2), 481–496 (2005)

    Article  MathSciNet  Google Scholar 

  20. Graves, L., Nomizu, K.: On sectional curvature of indefinite metric I. Math. Ann. 232, 267–272 (1978)

    Article  MathSciNet  Google Scholar 

  21. Gulbahar, M., Kilic, E., Celik, S.S.: Special proper pointwise slant surfaces of a locally product Riemannian manifold. Turkish. J. Math. 39, 884–899 (2015)

    Article  MathSciNet  Google Scholar 

  22. Gupta, G., Sachdeva, R., Kumar, R., Nagaich, R.K.: Pointwise slant lightlike submanifolds of indefinite Kaehler manifolds. Mediterr. J. Math. 15(3), 1–17 (2018)

    Article  MathSciNet  Google Scholar 

  23. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge (1972)

    Google Scholar 

  24. Haider, S. M. K., Advin, Thakur, M.: Hemi-slant lightlike submanifolds of indefinite Kaehler manifolds. J. Adv. Res. Dyn. Control Syst. 4, 10–22 (2012)

    Google Scholar 

  25. Lee, J.W., Sahin, B.: Pointwise slant submersions. Bull. Korean Math. Soc. 51, 1115–1126 (2014)

    Article  MathSciNet  Google Scholar 

  26. Leung, D.S., Nomizu, K.: The axiom of spheres in Riemannian geometry. J. Differential Geom. 5, 487–489 (1971)

    Article  MathSciNet  Google Scholar 

  27. Lotta, A.: Slant submanifolds in contact geometry. Bull. Math. Soc. Roumanie. 39, 183–198 (1996)

    MATH  Google Scholar 

  28. Maclane, S.: Geometrical Mechanics II, Lecture Notes. University of Chicago, USA (1968)

    Google Scholar 

  29. Maeda, S., Ohnita, Y., Udagawa, S.: On slant immersions into Kaehler manifolds. Kodai Math. J. 16, 205–219 (1993)

    Article  MathSciNet  Google Scholar 

  30. Matsumoto, K., Mihai, I., Tazawa, Y.: Ricci tensor of slant submanifolds in complex space forms. Kodai Math. J. 26, 85–94 (2003)

    Article  MathSciNet  Google Scholar 

  31. Nazaikinskii, V.E., Shatalov, V.E., Sternin, B.Y.: Contact Geometry and Linear Differential Equations. De Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin (1992)

    Google Scholar 

  32. Sachdeva, R., Kumar, R., Bhatia, S.S.: Totally umbilical hemi-slant lightlike submanifolds. New York J. Math. 21, 191–203 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Sachdeva, R., Kumar, R., Bhatia, S.S.: Totally umbilical slant lightlike submanifolds of indefinite Kaehler manifolds. Kyungpook Math. J. 57, 503–516 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Sachdeva, R., Kumar, R., Bhatia, S.S.: The axioms of indefinite hemi-slant planes and spheres. Lobachevskii J. Math. 38, 125–130 (2017)

    Article  MathSciNet  Google Scholar 

  35. Sahin, B.: Transversal lightlike submanifolds of indefinite Kaehler manifolds. An. Univ. Vest Timiş. Ser. Mat.-Inform. 44, 119–145 (2006)

    Google Scholar 

  36. Sahin, B.: Slant submanifolds of an almost product Riemannian manifold. J. Korean Math. Soc. 43(4), 717–732 (2006)

    Article  MathSciNet  Google Scholar 

  37. Sahin, B.: Slant lightlike submanifolds of indefinite Hermitian manifolds. Balkan J. Geom. Appl. 13, 107–119 (2008)

    MathSciNet  MATH  Google Scholar 

  38. Sahin, B.: Warped product submanifolds of Kaehler manifolds with a slant factor. Ann. Polon. Math. 95, 207–226 (2009)

    Article  MathSciNet  Google Scholar 

  39. Sahin, B.: Every totally umbilical proper slant submanifolds of a Kaehler manifold is totally geodesic. Result Math. 54, 167–172 (2009)

    Article  Google Scholar 

  40. Shukla, S.S., Yadav, A.: Semi-slant lightlike submanifolds of indefinite Kaehler manifolds. Rev. Un. Mat. Argentina 56(2), 21–37 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Shukla, S.S., Yadav, A.: Radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds. Mat. Vesnik 68(2), 119–129 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Shukla, S.S., Yadav, A.: Screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds. Novi Sad J. Math. 46(1), 147–158 (2016)

    Article  MathSciNet  Google Scholar 

  43. Shukla, S.S., Yadav, A.: Radical transversal screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds. SUT J. Math. 51(1), 31–47 (2015)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their sincere gratitude to the referees for their valuable suggestions, which definitely improved our manuscript.

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Authors and Affiliations

  1. Department of Basic and Applied Sciences, Punjabi University, Patiala, India

    Rashmi Sachdeva, Garima Gupta & Rakesh Kumar

  2. Department of Mathematics, University College, Ghanaur, Patiala, India

    Rachna Rani

  3. Department of Mathematics, Banaras Hindu University, Varanasi, India

    Akhilesh Yadav

Authors
  1. Rashmi Sachdeva
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  2. Garima Gupta
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  3. Rachna Rani
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  4. Rakesh Kumar
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  5. Akhilesh Yadav
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Correspondence to Rakesh Kumar .

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Editors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, USA

    Bang-Yen Chen

  2. Department of Mathematics, Jamia Millia Islamia, New Delhi, Delhi, India

    Mohammad Hasan Shahid

  3. President, King Khalid University, Abha, Saudi Arabia

    Falleh Al-Solamy

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Sachdeva, R., Gupta, G., Rani, R., Kumar, R., Yadav, A. (2022). Slant Lightlike Submanifolds of Indefinite Kaehler Manifolds and Their Warped Product Manifolds. In: Chen, BY., Shahid, M.H., Al-Solamy, F. (eds) Complex Geometry of Slant Submanifolds. Springer, Singapore. https://doi.org/10.1007/978-981-16-0021-0_12

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