## Taiwanese Journal of Mathematics

### EINSTEIN CONDITIONS FOR THE BASE SPACE OF ANTI-INVARIANT RIEMANNIAN SUBMERSIONS AND CLAIRAUT SUBMERSIONS

#### Abstract

In this paper, we study the geometry of anti-invariant Riemanniansubmersions from a Kähler manifold onto a Riemannian manifold. Wefirst determine the base space when the total space of ananti-invariant Riemannian submersion is Einstein and then weinvestigate new conditions for anti-invariant Riemannian submersionsto be Clairaut submersions. We also focus on the geometry of Clairaut anti-invariant submersions.

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 4 (2015), 1145-1160.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133693

Digital Object Identifier
doi:10.11650/tjm.19.2015.5283

Mathematical Reviews number (MathSciNet)
MR3384683

Zentralblatt MATH identifier
1357.53037

#### Citation

Lee, Jungchan; Park, JeongHyeong; Şahin, Bayram; Song, Dae-Yup. EINSTEIN CONDITIONS FOR THE BASE SPACE OF ANTI-INVARIANT RIEMANNIAN SUBMERSIONS AND CLAIRAUT SUBMERSIONS. Taiwanese J. Math. 19 (2015), no. 4, 1145--1160. doi:10.11650/tjm.19.2015.5283. https://projecteuclid.org/euclid.twjm/1499133693

#### References

• D. Allison, Lorentzian Clairaut submersions, Geom. Dedicata, 63 (1996), 309-319.
• K. Aso and S. Yorozu, A generalization of Clairaut's theorem and umbilic foliations, Nihonkai Math. J., 2 (1991), 139-153.
• R. L. Bishop, Clairaut submersions, differential geometry $($in Honor of Kentaro Yano$)$, Kinokuniya, Tokyo, 1972, pp. 21-31.
• R. H. Escobales Jr. and P. E. Parker, Geometric consequences of the normal curvature cohomology class in umbilic foliations, Indiana Univ. Math. J., 37 (1988), 389-408.
• M. Falcitelli, S. Ianus and A. M. Pastore, Riemannian Submersions and Related Topics, World Scientific, River Edge, NJ, 2004.
• P. Gilkey, J. Leahy and J. H. Park, Spectral geometry, Riemannian submersions and the Gromov-Lawson conjecture, Studies in Advanced Mathematics, CRC Press 1999.
• A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech., 16 (1967), 715-737.
• J. W. Lee, Anti-invariant $\xi^{\perp}$ Riemannian submersions from almost contact manifolds, Hacet. J. Math. Stat., 42 (2013), 231-241.
• B. O'Neill, The fundamental equations of a submersion, Mich. Math. J., 13 (1966), 458-469.
• B. \dSahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds, Cent. Eur. J. Math., 8 (2010), 437-447.
• B. \dSahin, Riemannian submersions from almost Hermitian manifolds, Taiwanese J. Math., 17 (2013), 629-659.
• H. M. Ta\dstan, On Lagrangian submersions, arXiv:1311.1676v2, Hacet. J. Math. Stat., to appear.
• B. Watson, Almost Hermitian submersions, J. Differential Geometry, 11 (1976), 147-165.
• K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore, 1984.