Harmonic Curvature, Harmonic Weyl Tensors and Related Geometric Flows on Complete Riemannian Manifolds

Author(s)	Dr. Indra Kant Jha
Country	India
Abstract	Let (Mn, g) be a complete Riemannian manifold. Harmonic curvature and harmonic Weyl curvature are important geometric conditions closely related to Einstein manifolds, Codazzi tensors and conformal geometry. A manifold is said to have harmonic curvature if the divergence of the Riemann curvature tensor vanishes, i.e. div R = 0. By the contracted second Bianchi identity, this condition is equivalent to the Ricci tensor being a Codazzi tensor. For , harmonic Weyl curvature is defined by div W = 0, which is equivalent to the vanishing of the Cotton tensor. In this paper, we discuss these curvature conditions on complete Riemannian manifolds and study their relationship with Einstein metrics and Ricci flow. Several basic theorems are proved, including the fact that harmonic curvature implies constant scalar curvature and that every Einstein manifold has harmonic curvature. The behavior of Einstein metrics under Ricci flow is also briefly examined.
Keywords	Harmonic curvature, Weyl tensor, Cotton tensor, Ricci tensor, Einstein manifold and Ricci flow etc.
Published In	Volume 2, Issue 12, December 2011
Published On	2011-12-03
DOI	https://doi.org/10.71097/IJTAS.v2.i12.1392

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