Conformal natural rescalings of Riemannian metrics and isometric embeddings of submanifolds: an obstruction from the Weyl tensor
Year: 2026
Type: preprint
Abstract: We consider the category $\mathrm{Riem}_n$ of connected Riemannian $n$-manifolds with open isometric embeddings as morphisms, and endofunctors of $\mathrm{Riem}_n$ that act as the identity on manifolds and smooth maps and replace each metric by another one. For $n \ge 4$, every smooth positive function $f$ of the squared Weyl norm $I_g = |W_g|^2$ defines a conformal natural rescaling $\Phi_f(g) = f(I_g)g$, and those for which $t \mapsto t f(t)^{-2}$ is a diffeomorphism of $[0,\infty)$ form an infinite-dimensional group of automorphisms of $\mathrm{Riem}_n$. We ask for which classes of lower-dimensional isometric embeddings, adjoined as morphisms to $\mathrm{Riem}_n$, compatibility with pullback forces $\Phi_f$ 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 $I_g$ 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 $(\Sigma^m, h)$ with $n = m+1 \ge 5$ we show that the set of values of $I_g$ at a point $p \in \Sigma$, over all isometric embeddings into Riemannian $n$-manifolds, is a half-line $[c(p), \infty)$ (or $(c(p), \infty)$), where $c(p)$ is the infimum over second fundamental forms $k$ of the squared Weyl norm of the tangential curvature $R^h_p - \tfrac12 k \owedge k$. Consequently, $\Phi_f$ is compatible with all isometric embeddings of $(\Sigma, h)$ if and only if $f$ is constant on $[\inf_p c(p), \infty)$, and an infinite-dimensional subgroup of automorphisms survives whenever $\inf_p c(p) > 0$. For $m = 4$ we compute $c$ in closed form, $c = 2\sum_i (w_i^+ - w_i^-)^2$ in terms of the ordered eigenvalues of $W^\pm$, so that $c = |W|^2/2$ for half-conformally-flat manifolds such as $\mathbb{CP}^2$, whereas $c = 0$ for $S^2 \times S^2$. For totally geodesic isometric embeddings in any codimension the threshold is $\inf_p |W^h_p|^2$. For three-dimensional hypersurfaces in dimension four the threshold vanishes and $f$ must be constant. We also show that conformal rescalings of the form $f(\mathrm{Scal}_g)\,g$ extending to all hypersurface embeddings, or to all totally geodesic embeddings of any dimension $1\le m\le n-1$, are forced to be constant multiples of the metric. code: MIT, manuscript/documentation: CC BY 4.0
Author: Miyata, Fumio
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225694226