Joon Sik Park, Kyung-Suk Moon
Abstract
In this paper, we get conditions for the natural projections of some product manifolds with varying metrics of two Riemannian manifolds to be harmonic, and necessary and sufficient conditions for some product manifolds with the harmonic natural projections of two Einstein manifolds to be Einstein manifolds.
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In this paper, we get conditions for the natural projections of some product manifolds with varying metrics of two Riemannian manifolds to be harmonic, and necessary and sufficient conditions for some product manifolds with the harmonic natural projections of two Einstein manifolds to be Einstein manifolds.
Key concepts: Einstein, Ricci-flat manifold, Product (mathematics), Pure mathematics, Mathematics, Harmonic, Natural (archaeology), Manifold (fluid mechanics)