Isometry and conformality of Riemannian manifolds to spheres (Q2716748)

Assume \((M,g)\) is a connected smooth Riemannian manifold of dimension \(n\geq 2\). Let \(G_{ji}\) and \(Z_{kjih}\) be respectively the zero trace part of the Ricci tensor and the concircular tensor, and NEWLINE\[NEWLINEW_{kjih}=aZ_{kjih}+b_1g_{kh}G_{ji}+b_2g_{ki} G_{jh}+b_3g_{ji}G_{kh}+b_4g_{jh}G_{ki}+b_5g_{kj}G_{ih}+b_6g_{ih}G_{kj},NEWLINE\]NEWLINE where \(a\) and the \(b_l\) (\(l=1,\dots,6\)) are constants. NEWLINENEWLINENEWLINEFor a smooth function \(\rho\) on \(M\), consider the connection with coefficients \({\overset{*}\Gamma}^h_{ji}=\Gamma^h_{ji}+\delta^h_i\rho_j\), where \(\rho_j=\partial_j\rho\) and the \(\Gamma^h_{ji}\) are the Christoffel symbols defined by \(g\). Thus, using the \({\overset{*}\Gamma}^h_{ji}\), one can define the tensors \({\overset{*} G}\), \({\overset{*} Z}\) and \({\overset{*} W}\) in the same way as for \(G\), \(Z\) and \(W\). NEWLINENEWLINENEWLINEThe author obtains some necessary and sufficient conditions involving \(|{\overset{*} G}|^2-|G|^2\) or \(|{\overset{*} W}|^2-|W|^2\) for a compact, orientable Riemannian manifold \(M\) to be conformal to a sphere.