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PreprintCC BY 4.0arXiv

Bi-Lagrangian and transverse Dirac structures induced on the Whitney sum

  • Bertuel Tangue Ndawa
  • Ferdinand Ngakeu
  • Nasser Saipele Nansidi
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Abstract

Let (M,ฯ‰,โ„ฑ1,โ„ฑ2) be a bi-Lagrangian manifold. For a distribution ๐’ŸโŠ† TM, we characterize when the subbundle N*t๐’Ÿ:=๐’ŸโŠ• N*๐’Ÿ of the Whitney sum W=TMโŠ• T*M is a Dirac structure for both the untwisted and (H,ฮธ)-twisted Courant brackets. Consequently, every bi-Lagrangian structure canonically determines a pair of transverse Dirac structures. the associated bi-Lagrangian connection, also called the Hess connection โˆ‡ induces a linear connection on W with respect to which both Dirac subbundles are parallel, thereby establishing a natural link between bi-Lagrangian geometry and Courant Dirac geometry. We also construct induced bi-Lagrangian structures on TM, T*M, and W, and study the relation between their affineness and that of the original structure. We also lift to these bundles the canonical action of the symplectomorphism group on bi-Lagrangian structures.

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Cite

BibTeXRIS
BibTeX
@misc{ndawa2026bilagrangian,
  title = {{Bi-Lagrangian and transverse Dirac structures induced on the Whitney sum}},
  author = {Bertuel Tangue Ndawa and Ferdinand Ngakeu and Nasser Saipele Nansidi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.11804},
  eprint = {2607.11804},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.11804},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Bi-Lagrangian and transverse Dirac structures induced on the Whitney sum
AU  - Bertuel Tangue Ndawa
AU  - Ferdinand Ngakeu
AU  - Nasser Saipele Nansidi
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2607.11804
UR  - https://arxiv.org/abs/2607.11804
AB  - Let (M,ฯ‰,โ„ฑ_1,โ„ฑ_2) be a bi-Lagrangian manifold. For a distribution ๐’ŸโŠ† TM, we characterize when the subbundle N^*t๐’Ÿ:=๐’ŸโŠ• N^*๐’Ÿ of the Whitney sum W=TMโŠ• T^*M is a Dirac structure for both the untwisted and (H,ฮธ)-twisted Courant brackets. Consequently, every bi-Lagrangian structure canonically determines a pair of transverse Dirac structures. the associated bi-Lagrangian connection, also called the Hess connection โˆ‡ induces a linear connection on W with respect to which both Dirac subbundles are parallel, thereby establishing a natural link between bi-Lagrangian geometry and Courant Dirac geometry. We also construct induced bi-Lagrangian structures on TM, T^*M, and W, and study the relation between their affineness and that of the original structure. We also lift to these bundles the canonical action of the symplectomorphism group on bi-Lagrangian structures.
ER  -