Conformal natural rescalings of Riemannian metrics and isometric embeddings of submanifolds: an obstruction from the Weyl tensor

We consider the category Riemₙ of connected Riemannian n-manifolds with open isometric embeddings as morphisms, and endofunctors of Riemₙ that act as the identity on manifolds and smooth maps and replace each metric by another one. For n ≥ 4, every smooth positive function f of the squared Weyl norm Ig = |Wg|² defines a conformal natural rescaling Φf(g) = f(Ig)g, and those for which t ↦ t f(t)⁻² is a diffeomorphism of [0,∞) form an infinite-dimensional group of automorphisms of Riemₙ. We ask for which classes of lower-dimensional isometric embeddings, adjoined as morphisms to Riemₙ, compatibility with pullback forces Φf to be a constant multiple of the metric. Because the squared Weyl norm of the ambient curvature is bounded below by that of its tangential part, which is determined by the Gauss equation, the values of Ig attainable along isometric embeddings of a single manifold are bounded below, and the answer depends on the intrinsic geometry of that manifold: for a fixed Riemannian manifold (Σᵐ, h) with n = m+1 ≥ 5 we show that the set of values of Ig at a point p ∈ Σ, over all isometric embeddings into Riemannian n-manifolds, is a half-line [c(p), ∞) (or (c(p), ∞)), where $c(p)$ is the infimum over second fundamental forms k of the squared Weyl norm of the tangential curvature Rʰₚ - 12 k k. Consequently, Φf is compatible with all isometric embeddings of (Σ, h) if and only if f is constant on [infₚ c(p), ∞), and an infinite-dimensional subgroup of automorphisms survives whenever infₚ c(p) > 0. For $m = 4$ we compute c in closed form, c = 2∑ᵢ (wᵢ⁺ - wᵢ⁻)² in terms of the ordered eigenvalues of W^±, so that c = |W|²/2 for half-conformally-flat manifolds such as CP², whereas $c = 0$ for S² × S². For totally geodesic isometric embeddings in any codimension the threshold is infₚ |Wʰₚ|². For three-dimensional hypersurfaces in dimension four the threshold vanishes and f must be constant. We also show that conformal rescalings of the form f(Scalg)\,g extending to all hypersurface embeddings, or to all totally geodesic embeddings of any dimension 1≤ m≤ n-1, are forced to be constant multiples of the metric. code: MIT, manuscript/documentation: CC BY 4.0