On doubly warped products

Year 2020, Volume: 69 Issue: 2, 1329 - 1335, 31.12.2020

Sibel Gerdan Aydın , Hakan Mete Taştan

https://doi.org/10.31801/cfsuasmas.635048

Cited By: 2

Abstract

We give a new characterization for doubly warped products by using the geometry of their canonical foliations intersecting perpendicularly. We also give a necessary and sufficient condition for a doubly warped product to be a warped or a direct product. As a result, we prove the non-existence of Einstein proper doubly warped product pseudo-Riemannian manifold of dimension grater or equal than 4.                                                                                                    .

.

Keywords

Doubly warped product, warped product, direct product, Einstein manifold

References

• Y. Agaoka, I-B. Kim, D.J. Yeom,On doubly warped product manifolds, Mem. Fac. Integrated Arts and Sci., Hiroshimo Univ., ser. IV. 24 (1998), 1-10.
• R.L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Mat. Soc. 145(1) (1969), 1-49.
• R.A. Blumenthal and J.J. Hebda, An analogue of the holonomy bundle for a foliated manifold, Tohoku Math. J. 40(2) (1988), 189-197.
• B.Y. Chen, Geometry of submanifolds and its applicaitons, Science University of Tokyo, Tokyo, 1981.
• B.Y. Chen, Differential geometry of warped product manifolds and submanifolds, World Scientific, 2017.
• P. E. Ehrlich, Metric deformations of Ricci and sectional curvature on compactRiemannian manifolds, Ph.D. dissertation, SUNY, Stony Brook, N.Y., 1974.
• M. Fernandez Lopez , E. Garcia Rio , D.N. Küpeli, B. Ünal, A curvature condition for a twisted product to be a warped product, Manuscripta Math. 106 (2001), 213-217.
• P. Gupta, On compact Einstein doubly warped product manifolds, Tamkang J. Math. 49(4) (2018), 267-275. doi:10.5556/j.tkjm.49.2018.2605.
• G.I. Kruchkovich, On semi-reducible Riemannian spaces (in Russian), Dokl. Akad. Nauk SSSR 115 (1957), 862-865.
• B. Olea, Doubly warped product structures on semi-Riemannian manifolds, Ph.D. thesis, Universty of Malaga, 2009.
• B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, San Diego, 1983.
• R. Ponge and H. Reckziegel, Twisted products in pseudo- Riemannian geometry, Geom. Dedicata 48 (1993), 15-25.
• B. Ünal, Doubly warped products, Ph.D. thesis, Universty of Missouri-Columbia, 2000.
• K. Yano, M. Kon, Structures on Manifolds, Singapore: World Scientific, 1984.