Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 062, 36 pages      arXiv:1703.03851

Lie Algebroid Invariants for Subgeometry

Anthony D. Blaom
Waiheke Island, New Zealand

Received November 15, 2017, in final form June 13, 2018; Published online June 18, 2018

We investigate the infinitesimal invariants of an immersed submanifold $\Sigma $ of a Klein geometry $M\cong G/H$, and in particular an invariant filtration of Lie algebroids over $\Sigma $. The invariants are derived from the logarithmic derivative of the immersion of $\Sigma $ into $M$, a complete invariant introduced in the companion article, A characterization of smooth maps into a homogeneous space. Applications of the Lie algebroid approach to subgeometry include a new interpretation of Cartan's method of moving frames and a novel proof of the fundamental theorem of hypersurfaces in Euclidean, elliptic and hyperbolic geometry.

Key words: subgeometry; Lie algebroids; Cartan geometry; Klein geometry; differential invariants.

pdf (655 kb)   tex (124 kb)


  1. Armstrong S., Note on pre-Courant algebroid structures for parabolic geometries, arXiv:0709.0919.
  2. Blaom A.D., Geometric structures as deformed infinitesimal symmetries, Trans. Amer. Math. Soc. 358 (2006), 3651-3671, math.DG/0404313.
  3. Blaom A.D., Lie algebroids and Cartan's method of equivalence, Trans. Amer. Math. Soc. 364 (2012), 3071-3135, math.DG/0509071.
  4. Blaom A.D., The infinitesimalization and reconstruction of locally homogeneous manifolds, SIGMA 9 (2013), 074, 19 pages, arXiv:1304.7838.
  5. Blaom A.D., Cartan connections on Lie groupoids and their integrability, SIGMA 12 (2016), 114, 26 pages, arXiv:1605.04365.
  6. Blaom A.D., Pseudogroups via pseudoactions: unifying local, global, and infinitesimal symmetry, J. Lie Theory 26 (2016), 535-565, arXiv:1410.6981.
  7. Blaom A.D., A characterization of smooth maps into a homogeneous space, arXiv:1702.02717.
  8. Blaom A.D., The Lie algebroid associated with a hypersurface, arXiv:1702.03452.
  9. Burstall F.E., Calderbank D.M.J., Submanifold geometry in generalized flag manifolds, Rend. Circ. Mat. Palermo (2) Suppl. 72 (2004), 13-41.
  10. Burstall F.E., Calderbank D.M.J., Conformal submanifold geometry I-III, arXiv:1006.5700.
  11. Cannas da Silva A., Weinstein A., Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, Vol. 10, Amer. Math. Soc., Providence, RI, Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999.
  12. Čap A., Gover A.R., Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2002), 1511-1548.
  13. Čap A., Slovák J., Parabolic geometries. I. Background and general theory, Mathematical Surveys and Monographs, Vol. 154, Amer. Math. Soc., Providence, RI, 2009.
  14. Crainic M., Fernandes R.L., Lectures on integrability of Lie brackets, in Lectures on Poisson Geometry, Geom. Topol. Monogr., Vol. 17, Geom. Topol. Publ., Coventry, 2011, 1-107, math.DG/0611259.
  15. Dufour J.-P., Zung N.T., Poisson structures and their normal forms, Progress in Mathematics, Vol. 242, Birkhäuser Verlag, Basel, 2005.
  16. Fels M., Olver P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
  17. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  18. Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.
  19. Olver P.J., A survey of moving frames, in Computer Algebra and Geometric Algebra with Applications (6th International Workshop, IWMM 2004, Shanghai, China, May 19-21, 2004 and International Workshop, GIAE 2004, Xian, China, May 24-28, 2004), Geom. Topol. Monogr., Vol. 17, Editors H. Li, P.J. Olver, G. Sommer, Springer, Berlin - Heidelberg, 2005, 105-138.
  20. Sharpe R.W., Differential geometry. Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, Vol. 166, Springer-Verlag, New York, 1997.
  21. Xu X., Twisted Courant algebroids and coisotropic Cartan geometries, J. Geom. Phys. 82 (2014), 124-131, arXiv:1206.2282.

Previous article  Next article   Contents of Volume 14 (2018)