Topology of isometric classes and flows of geometric structures
Abstract
We study the topology of isometric classes of tensorial H-structures for closed and connected subgroups HleqslantSO(n). The map assigning an H-structure its induced metric admits a global trivialization; consequently, the space of H-structures is homotopy equivalent to any fixed isometric class. On flat tori, these classes are mapping spaces into SO(n)/H. We give examples with infinitely many connected components for G2, Spin(7), and almost Hermitian structures, determine the number of components for SU(m)-structures on flat tori, and study the corresponding moduli spaces. For closed manifolds of dimension n>2, the intrinsic torsion energy on the full space of H-structures has infimum zero on every nonempty path component, and its critical points are precisely the torsion-free structures. In fixed isometric classes, we contrast the concentration phenomena underlying finite-time singularities with cohomological classes admitting positive energy lower bounds and harmonic representatives. Finally, we establish a lifting principle for metric-dependent flows, derive an evolution identity for isometric flows, and extend harmonic-flow estimates to general H-structures.
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BibTeX
@misc{fadel2026topology,
title = {{Topology of isometric classes and flows of geometric structures}},
author = {Daniel Fadel and Eric Loubeau},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.12031},
eprint = {2606.12031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12031},
}
RIS
TY - UNPB
TI - Topology of isometric classes and flows of geometric structures
AU - Daniel Fadel
AU - Eric Loubeau
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2606.12031
UR - https://arxiv.org/abs/2606.12031
AB - We study the topology of isometric classes of tensorial H-structures for closed and connected subgroups HleqslantSO(n). The map assigning an H-structure its induced metric admits a global trivialization; consequently, the space of H-structures is homotopy equivalent to any fixed isometric class. On flat tori, these classes are mapping spaces into SO(n)/H. We give examples with infinitely many connected components for G_2, Spin(7), and almost Hermitian structures, determine the number of components for SU(m)-structures on flat tori, and study the corresponding moduli spaces. For closed manifolds of dimension n>2, the intrinsic torsion energy on the full space of H-structures has infimum zero on every nonempty path component, and its critical points are precisely the torsion-free structures. In fixed isometric classes, we contrast the concentration phenomena underlying finite-time singularities with cohomological classes admitting positive energy lower bounds and harmonic representatives. Finally, we establish a lifting principle for metric-dependent flows, derive an evolution identity for isometric flows, and extend harmonic-flow estimates to general H-structures.
ER -