Weyl tensors, strongly regular graphs, multiplicative characters, and a quadratic matrix equation (Q6568814)

In this paper, the authors study solutions of a quadratic matrix equation arising in Riemannian geometry. The Ricci flow is a solution to the PDE \(\frac{\partial}{\partial t} \langle -,- \rangle + \langle \mathrm{Ric}(R)-,- \rangle=0\) for a time-dependent Riemannian metric \(\langle -,- \rangle\) with Ricci tensor \(\mathrm{Ric}(R)\) on a manifold \(M\) and the curvature tensor \(R\) satisfies the partial differential equation \(\frac{\partial}{\partial t}R=\Delta R+2\big (R^{2}+R^{\#} \big)\). Hamilton's maximum principle asserts that certain positivity assumptions are preserved by the Ricci flow if they are preserved by the ordinary differential equation \(\frac{d}{dt}R=R^{2}+R^{\#}\). It is then of interest to understand solutions \((R, \theta)\) of the quadratic equation \(\theta R=R^{2}+R^{\#}\): such solutions can be viewed as obstructions to an evolution towards a metric with constant positive sectional curvature.\N\NThe authors' main result is the following (Theorem 1.1): Let \(V\) be a Euclidean vector space with an orthonormal basis \(e_{1}, \ldots, e_{n}\), with \(n \geq 4\). Then there exists an algebraic curvature tensor \(R\) which is Einstein, such that the \(e_{i} \wedge e_{j}\) with \(i < j\) are eigenvectors of \(R\), and a real number \(\theta \geq 0\) with\N\[\N\theta R=R^{2}+R^{\#},\N\]\Nand \(R\) is not a multiple of the identity.\N\NSuch solutions \(R\), where the eigenvalue spectrum of \(R\) consists of precisely two real numbers, are (up to scaling) in one-to-one correspondence with strongly regular graphs \(\Gamma\). If both \(\Gamma\) and its complementary graph \(\Gamma^{c}\) are connected, then \(R\) is not the curvature tensor of a product of spheres. In this case the dimension \(n\) is the number of vertices of the graph. Other nontrivial solutions in dimension \(n\) can be constructed from multiplicative characters of finite fields \(\mathbb{F}_{q}\), where \(n=q\) is a prime power with \(q \equiv 1 \mod m\) and \(m \in \{3,4,8 \}\).