ON WARPED PRODUCT MANIFOLDS - CONFORMAL FLATNESS AND CONSTANT SCALAR CURVATURE PROBLEM

Authors

  • KWANG-WU YANG Yuan-Ze Institute of Technology, Energy Technology Center, 135 Yuan-Tung Road., Ne-Li, Taoyuan Shian, Taiwan 32026, R.O.C.

DOI:

https://doi.org/10.5556/j.tkjm.29.1998.4272

Keywords:

Warped porduct, conformal flat, scalar curvature

Abstract

In this paper, we study some geometric properties on doubly or singly warped­ product manifolds. In general, on a fixed topological product manifold, the problem for finding warped-product metrics satisfying certain curvature conditions are finally reduced to find positive solutions of linear or non-linear differential equations. Here, we are mainly interested in the following problems on essentially warped-product manifolds: one is the sufficient and necessary conditions for conformal flatness, and the other is to find warped-product metrics so that their scalar curvatures are contants.

References

A. Aubin., "Nonlinear analysis on manifolds. Monge-ampere equations," A Series of Com­ prehensive Studies in Mathematics, Vol. 252, Spring-Verlag Berlin, 1982.

R. L. Bishop and B. O'Neill., "Manifolds of negative curvature," Trans. Amer . Math. Soc. 145(1969), 1-49.

J. P.B ourguignon., "Les varietes de dimension 4 asignature non nulle dont la courbure est harmonique sont d'Einstein," Invent. Math., 63(1981), 263~286.

M. C. Chaki and B. Gupta., "On conformally symmetric spaces," Indian J. Math., 5(1963), 113-122.

A. Derdzinski., "A classification of certam compact nemannian manifolds with harmonic curvature and non-parallel ricci tensor," Math. z., 172(1980), 273-280.

A. Derdzinski. "On compact Riemanman manifolds with harmonic curvature," Math. Ann., 259(1982), 145-152.

A. Dedzinski., "On conformally symmetric ricci-recurrent manifolds with abelian funda­ mental group," Tensor, N. S., 34(1980), 21-29.

A. Derdzinski and W. Rater. "On conformally symmetric manifolds with metrics of indices 0 and 1," Tensor , N. S., 31(1977), 255-259.

R. Deszcz., "On four-dimensional Riemannian warped product manifolds satisfying certain pseudo-symmetry curvature conditions," Colloquium Math., 62(199,1), 103-120.

F.Dobarro and E. Lami-Dozo., "Scalar curvature and warped products of Riemann manifolds," Trans. Amer. Math. Soc., 303(1987), 161-168.

N. Ejiri., "Some compact hypersurfaces of constant scalar curvature in a sphere," J. Geom., 19(1982), 197-199.

A. Gray., "Einstein-like manifolds which are not Einstein," Geom. Dedicata, 7(1978) , 259-280.

M. Hotlos., "Some theorems on doubly warped products," Demonstrdtio Math., 23(1990), 39-58.

S. Kobayashi and K. Nomizu., Foundations of Differential Geometry, Vol. !(1963), 11(1969), Interscience, Wiley, New York.

Y. Ogawa., "On isometric immersions of conformally flat Riemanman spaces with negative sectional curvature " Natur . Sci. Rep. Ochanomizu Univ., 31(1980), 13-21.

W. Roter., "On conformally symmetric ricci-recurrent space," Colloquium Math., 31(1974) 87-96.

K. Yano., Integral Formulas in Riemannian Geometry, Marcel Dekker, Inc. New York, 1970.

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Published

1998-09-01

How to Cite

YANG, K.-W. (1998). ON WARPED PRODUCT MANIFOLDS - CONFORMAL FLATNESS AND CONSTANT SCALAR CURVATURE PROBLEM. Tamkang Journal of Mathematics, 29(3), 203-221. https://doi.org/10.5556/j.tkjm.29.1998.4272

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Papers