Conformal semi-invariant Riemannian maps to Sasakian manifolds

Abstract

The idea of conformal semi-invariant Riemannian maps to almost Hermitian manifolds was first put forward by Şahin and Akyol in [3]. In this paper, we expand this idea to Sasakian manifolds which are almost contact metric manifolds. Hereby, we present conformal semi-invariant Riemannian maps from Riemannian manifolds to Sasakian manifolds. Then, we prepare a illustrative example and investigate the geometry of the leaves of $D_1$, $D_2$, $\overline{D}_1$ and $\overline{D}_2$. We find necessary and sufficient conditions for conformal semi-invariant Riemannian maps to be totally geodesic. Also, we investigate the harmonicity of such maps.

Keywords

• Semi-invariant Riemannian map
• conformal semi-invariant Riemannian map
• Sasakian manifolds

References

1. Akyol, M. A., Şahin, B., Conformal slant Riemannian maps to Kahler manifolds, Tokyo J. Math., 42(1) (2019), 225–237. DOI: 10.3836/tjm/1502179277
2. Akyol, M. A., Şahin, B., Conformal anti-invariant Riemannian maps to Kaehler manifolds, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 80(4) (2018), 187–198.
3. Akyol, M. A., Şahin, B., Conformal semi-invariant Riemannian maps to K¨ahler manifolds, Revista de la Uni´on Matematica Argentina, 60(2) (2019), 459-468. https://doi.org/10.33044/revuma.v60n2a12
4. Blair, D. E., Contact Manifolds in Riemannian Geometry; Lecture Notes in Math 509; Springer: Berlin/Heidelberg, Germany, New York, NY, USA, 1976.
5. Cabrerizo, J. L., Carriazo, A., Fernandez, L. M., Fernandez, M., Slant submanifolds in Sasakian manifolds, Glasgow Math. J., 42(1) (2000), 125-138. DOI: https://doi.org/10.1017/S0017089500010156
6. Fischer, A. E., Riemannian maps between Riemannian manifolds, Contemp. Math., 132 (1992), 331–366.
7. Gündüzalp, Y., Akyol, M. A., Remarks on conformal anti-invariant Riemannian maps to cosymplectic manifolds, Hacettepe J. Math. Stat., (2021), 1–9. DOI : 10.15672/hujms.677910
8. Jaiswal, J. P., Harmonic maps on Sasakian manifolds, J. Geom., 104(2) (2013), 309–315. DOI 10.1007/s00022-013-0158-2

9. Kumar, S., Prasad, R., Kumar, S., Clairaut semi-invariant Riemannian maps from almost Hermitian manifolds, Turk. J. Math., 46(4) (2022), 1193–1209. DOI : 10.55730/1300-0098.3151
10. Meena, K., Şahin, B., Shah, H. M., Riemannian warped product maps, Results Math., 79(2) (2024), 1–30. https://doi.org/10.1007/s00025-023-02084-1
11. Meena, K., Zawadzki, T., Clairaut conformal submersions, Bull. Malays. Math. Sci. Soc., 47(4) (2024), 1–22. https://doi.org/10.1007/s40840-024-01697-1
12. Meena, K., Shah, H. M., Şahin, B., Geometry of Clairaut conformal Riemannian maps, J. Austr. Math. Soc., (2024), 1-39. doi: https://doi.org/10.1017/S1446788724000090
13. Meena, K., Yadav, A., Clairaut Riemannian maps, Turk. J. Math., 47(2) (2023), 794–815. DOI : 10.55730/1300-0098.3394
14. Nore, T., Second fundamental form of a map, Annali di Matematica pura ed applicata, 146 (1986), 281–310. https://doi.org/10.1007/BF01762368
15. Pandey, B., Jaiswal, J. P., Ojha, R. H., Necessary and sufficient conditions for the Riemannian map to be a harmonic map on cosymplectic manifolds, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 85(2) (2015), 265–268. https://doi.org/10.1007/s40010-015-0203-7
16. Prasad, R., Pandey, S., Slant Riemannian maps from an almost contact manifold, Filomat, 31(13) (2017), 3999–4007. https://doi.org/10.2298/FIL1713999P
17. Prasad, R., Kumar, S., Slant Riemannian maps from Kenmotsu manifolds into Riemannian manifolds, Glob. J. Pure Appl. Math., 13 (2017), 1143–1155. DOI:10.37622/000000
18. Polat, M., Meena, K., Clairaut semi-invariant Riemannian maps to Kahler manifolds, Mediterr. J. Math., 21(3) (2024), 1–19. https://doi.org/10.1007/s00009-024-02666-5
19. Ponge, R., Reckziegel, H., Twisted products in pseudo-Riemannian geometry, Geometriae Dedicata, 48(1) (1993), 15-25. https://doi.org/10.1007/BF01265674
20. Şahin, B., Invariant and anti-invariant Riemannian maps to Kahler manifolds, Int. J. Geom. Methods Mod. Phys., 7(3) (2010), 1–19. https://doi.org/10.1142/S0219887810004324
21. Şahin, B., Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and Their Applications, Elsevier, Academic Press, 2017.
22. Şahin, B., Conformal Riemannian maps between Riemannian manifolds, their harmonicity and decomposition theorems, Acta Appl. Math., 109(3) (2010), 829–847. https://doi.org/10.1007/s10440-008-9348-6
23. Şahin, B., Yanan, Ş., Conformal Riemannian maps from almost Hermitian manifolds, Turkish J. Math., 42(5) (2018), 2436–2451. DOI : 10.3906/mat-1711-34
24. Şahin, B., Semi-invariant Riemannian maps from almost Hermitian manifolds, Indag. Math., 23 (2012), 80–94. https://doi.org/10.1016/j.indag.2011.10.003
25. Şahin, B., Slant Riemannian maps from almost Hermitian manifolds, Quaest. Math., 36 (2013), 449–461. https://doi.org/10.2989/16073606.2013.779968
26. Şahin, B., Circles along a Riemannian map and Clairaut Riemannian maps, Bull. Korean Math. Soc., 54(1) (2017), 253–264. https://doi.org/10.4134/BKMS.b160082
27. Yadav, A., Meena, K., Riemannian maps whose total manifolds admit a Ricci soliton, J. Geom. Phys., 168 (2021), 1–13. https://doi.org/10.1016/j.geomphys.2021.104317
28. Yadav, A., Meena, K., Clairaut Riemannian maps whose total manifolds admit a Ricci soliton, Int. J. Geom. Methods Mod. Phys., 19(2) (2022), 1–17. https://doi.org/10.1142/S0219887822500244
29. Yadav, A., Meena, K., Riemannian maps whose base manifolds admit a Ricci soliton, Publ. Math. Debrecen, 103(1-2) (2023), 115–139. https://doi.org/10.48550/arXiv.2107.01049
30. Yadav, A., Meena, K., Clairaut invariant Riemannian maps with Kahler structure, Turk. J. Math., 46(3) (2022), 1020–1035. DOI : 10.55730/1300-0098.3139
31. Yadav, A., Meena, K., Clairaut anti-invariant Riemannian maps from Kahler manifolds, Mediterr. J. Math., 19(2) (2022), 1–19. https://doi.org/10.1007/s00009-022-02018-1