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PreprintCC BY 4.0arXiv

A Cartan-geometrical perspective on torsion and non-metricity

  • Damianos Iosifidis
  • Tomi Koivisto
  • Wit Krośnicki
  • Tom Zlosnik
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Abstract

Élie Cartan established that the metric and intrinsic curvature of a D dimensional embedded manifold M could be determined by tracing the response of another surface N of the same dimensionality, as it is rolled without slipping and twisting on M. In the context of spacetime geometry, this construction underpins the MacDowell-Mansouri formulation of General Relativity. We consider extensions of this framework that correspond to rolling of a shape with twisting and with shape evolution; it is shown that these naturally describe torsion and non-metricity respectively. Cartan-geometric formulations of teleparallel gravity and symmetric teleparallelism are discussed in addition to further manifestations of non-metricity in first-order formulations of gravity.

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Cite

BibTeXRIS
BibTeX
@misc{iosifidis2026cartangeometrical,
  title = {{A Cartan-geometrical perspective on torsion and non-metricity}},
  author = {Damianos Iosifidis and Tomi Koivisto and Wit Krośnicki and Tom Zlosnik},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30093},
  eprint = {2609.30093},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30093},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Cartan-geometrical perspective on torsion and non-metricity
AU  - Damianos Iosifidis
AU  - Tomi Koivisto
AU  - Wit Krośnicki
AU  - Tom Zlosnik
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30093
UR  - https://arxiv.org/abs/2609.30093
AB  - Élie Cartan established that the metric and intrinsic curvature of a D dimensional embedded manifold M could be determined by tracing the response of another surface N of the same dimensionality, as it is rolled without slipping and twisting on M. In the context of spacetime geometry, this construction underpins the MacDowell-Mansouri formulation of General Relativity. We consider extensions of this framework that correspond to rolling of a shape with twisting and with shape evolution; it is shown that these naturally describe torsion and non-metricity respectively. Cartan-geometric formulations of teleparallel gravity and symmetric teleparallelism are discussed in addition to further manifestations of non-metricity in first-order formulations of gravity.
ER  -