Generic properties of critical points of the Weyl tensor (Q507454)

The Riemann curvature tensor of a Riemannian manifold \((M^n,g)\) decomposes into three parts corresponding to the irreducible modules for the action of the orthogonal group \(\mathsf{O}(n)\) on the space of symmetric endomorphisms of \(\bigwedge^2(T_p M)\): the scalar curvature, the traceless Ricci tensor, and the Weyl tensor. The latter is known to be a conformal invariant, and hence relevant for questions such as compactness of solutions to the Yamabe problem. Motivated by this, the authors prove that for a smooth manifold \(M^n\), within the set \(\mathcal{D}^k\) of \(C^k\)-Riemannian metrics on \(M\) whose Weyl tensor is nowhere vanishing, those for which every critical point of the norm squared of the Weyl tensor \(\mathcal{W}_g = | \text{Weyl} |_g^2\) is nondegenerate form an open dense subset. The proof relies on a transversality condition for the derivative of \(\mathcal{W}\), which the authors check via an explicit computation.