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ON STABILITY OF EINSTEIN WARPED PRODUCT MANIFOLDS

  • Pyo, Yong-Soo (Department of Applied Mathematics, Pukyong National University) ;
  • Kim, Hyun-Woong (Department of Applied Mathematics, Pukyong National University) ;
  • Park, Joon-Sik (Department of Mathematics, Pusan University of Foreign Studies)
  • Received : 2009.09.24
  • Accepted : 2010.03.12
  • Published : 2010.03.25

Abstract

Let (B, $\check{g}$) and (N, $\hat{g}$) be Einstein manifolds. Then, we get a complete (necessary and sufficient) condition for the warped product manifold $B\;{\times}_f\;N\;:=\;(B\;{\times}\;N,\;\check{g}\;+\;f{\hat{g}}$) to be Einstein, and obtain a complete condition for the Einstein warped product manifold $B\;{\times}_f\;N$ to be weakly stable. Moreover, we get a complete condition for the map i : ($B,\;\check{g})\;{\times}\;(N,\;\hat{g})\;{\rightarrow}\;B\;{\times}_f\;N$, which is the identity map as a map, to be harmonic. Under the assumption that i is harmonic, we obtain a complete condition for $B\;{\times}_f\;N$ to be Einstein.

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References

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