Bach flows of product manifolds (Q2920426)

The Bach tensor arises in the theory of conformal gravity and on the unwarped \((2,2)\) product manifolds it is completely determined by the derivatives of the Ricci scalars of the individual two-dimensional manifolds of the product. The present paper starts by analyzing the Bach flow on typical examples of such \((2,2)\) manifolds: \(\mathbb{S}^2 \times \mathbb{S}^2\) and \(\mathbb{R}^2 \times \mathbb{S}^2\). On the general unwarped \((2,2)\) product manifolds, the authors obtain a system of equations for the fixed point of the flow which can be solved numerically. Furthermore, the Bach flow is shown to reduce to a first order dynamical system for a special class of \(4\)-dimensional warped manifolds. Throughout the paper, differences and similarities with the Ricci flow are pointed out.