Topology of isometric classes and flows of geometric structures

Abstract

We revisit flows of tensorial $H$-structures for closed and connected Lie subgroups $H\leqslant\mathrm{SO}(n)$, focusing on the topology of isometric classes. We prove that the natural map assigning to an $H$-structure its induced Riemannian metric is surjective and satisfies a parametric homotopy lifting property. Since the space of Riemannian metrics is contractible, the full space of $H$-structures is homotopy equivalent to any fixed isometric class. For parallelizable manifolds, especially flat tori, these classes reduce to mapping spaces into $\mathrm{SO}(n)/H$. We discuss almost Hermitian, $\mathrm{SU}(m)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$ structures on flat tori, showing that their isometric classes and moduli modulo orientation-preserving diffeomorphisms may have infinitely many connected components. We relate this topology to the variational theory of the intrinsic torsion energy. On the unrestricted space of $H$-structures, the functional is scale-degenerate in dimensions $n>2$: its infimum is zero on every nonempty path component, and its only critical points are torsion-free structures. Inside fixed isometric classes this homothetic escape direction is absent. We reinterpret finite-time singularity formation as concentration in nontrivial isometric homotopy classes with zero energy infimum, and contrast this with cohomological classes, such as $\mathrm{U}(3)$-structures on the flat $6$-torus, which have positive lower bounds and admit smooth harmonic representatives from holomorphic maps into $\mathbb{CP}^3$. Finally, we revisit analytical aspects of our earlier work: we prove a lifting principle for metric-dependent flows, reinterpret the Ricci $H$-flow, derive a general evolution identity for isometric flows, and extend the harmonic-flow theory beyond the original structural assumptions.