Clairaut and Einstein conditions for locally conformal Kaehler submersions

Yıl 2023, , 28 - 39, 21.06.2023

Beran Pirinççi , Çağrıhan Çimen , Deniz Ulusoy

https://doi.org/10.26650/ijmath.2023.00003

Öz

In the present paper, we study Clairaut submersions and Einstein conditions whose total manifolds are locally conformal Kaehler manifolds. We first give a necessary and sufficient condition for a curve to be geodesic on total manifold of a locally conformal Kaehler submersion. Then, we investigate conditions for a locally conformal Kaehler submersion to be a Clairaut submersion.We find the Ricci and scalar curvature formulas between any fiber of the total manifold and the base manifold of a locally conformal Kaehler submersion and give necessary and sufficient conditions for the total manifold of a locally conformal Kaehler submersion to be Einstein. Finally, we obtain some formulas for sectional and holomorphic sectional curvatures for a locally conformal Kaehler submersion.

Anahtar Kelimeler

Riemannian submersion, almost Hermitian submersion, locally conformal Kaehler submersion, Clairaut submersion, Einstein manifold

Kaynakça

• Allison, D., 1996, Lorentzian Clairaut submersions, Geom. Dedicata, 63(3), 309-319. google scholar
• Bishop, R.L., 1972, Clairaut submersions, differential geometry (in Honor of Kentaro Yano), Tokyo, Kinokuniya, 21-31. google scholar
• Çimen Ç., Pirinççi B., Taştan H.M., Ulusoy D., 2023, On locally conformal Kaehler submersions (Submitted). google scholar
• Dragomir, S., Ornea, L., 1998, Locally conformal Kahler geometry, Birkhauser: Boston, Basel, Berlin. google scholar
• Falcitelli, M., Lanus, S., Pastore, A.M., 2004, Riemannian Submersion and Related Topics, World Scientific Publishing Co. Pte. Ltd.: Singapore Gray, A., 1967, Pseudo-Riemannian Almost Product Manifolds and Submersions, Journal of Mathematics and Mechanics, (16)7, 715-737. google scholar
• Gündüzalp, Y., 2020, Clairaut anti-invariant submersions from locally product Riemannian manifolds, Beitr. Algebra Geom., 61, 605-614. google scholar
• Lee, J., Park, J.H., Şahin, B., Song, D.Y., 2015, Einstein conditions for the base space of anti-invariant Riemannian submersions and Clairaut submersions, Taiwanese J. Math., 19(4), 1145-1160. google scholar
• Marrero, J.C., Rocha, J., 1994, Locally conformal Kaehler submersions, Geom. Dedicata, 52(3), 271-289. google scholar
• O’Neill, B., 1966, The fundamental equations of a submersion, Mich. Math. J., 13, 458-469, google scholar
• O’Neill, B. 1967 Submersions and Geodesics, Duke Math. J., 34(2), 363-373. google scholar
• Şahin, B., 2017, Riemannian submersions, Riemannian maps in Hermitian geometry, and their application, Elsiever. google scholar
• Taştan, H.M., Gerdan S., 2016, Clairaut anti-invariant submersions from Sasakian and Kenmotsu manifolds, Mediterr. J. Math., 14(6), 234-251. google scholar
• Taştan, H.M., Gerdan Aydın, S., 2019, Clairaut anti-invariant submersions from cosymplectic manifolds, Honam Math. J., 41(4), 707-724. google scholar
• Vaisman, I., 1980, Some curvature properties of locally conformal Kaehler manifolds, Trans. Amer. Math. Soc., 259, 439-447. google scholar
• Watson, B., 1976, Almost Hermitian submersions, J. Differ. Geom., 11(1), 147-165. google scholar