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An intrinsic and exterior form of the Bianchi identities

Thoan Do

Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086, Australia

E-mail Address: dtkthoan@ctu.edu.vn

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and

Geoff Prince

Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086, Australia

The Australian Mathematical Sciences Institute, c/o The University of Melbourne, Parkville, Victoria 3010, Australia

E-mail Address: geoff@amsi.org.au

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https://doi.org/10.1142/S0219887817500013Cited by:0 (Source: Crossref)

Abstract

We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms of arbitrary degree. We demonstrate the relationship with both the conventional vector version of the Bianchi identities and to the exterior covariant derivative approach. Contact manifolds, codimension one foliations and the Cartan form of classical mechanics are studied as examples of its flexibility and utility.

Keywords:

AMSC: 53B05, 53C05, 58A10

References

1. A. Bejancu and H. R. Farran, Foliations and Geometric Structures (Springer, 2006). Google Scholar
2. S. S. Chern and R. S. Hamilton, On Riemannian metrics adapted to three-dimensional contact manifolds, Lecture Notes in Mathematics, Vol. 1111 (Springer-Verlag, 1985), pp. 279–305. Crossref, Google Scholar
3. M. Crampin and F. A. E. Pirani, Applicable Differential Geometry (Cambridge University Press, 1986). Google Scholar
4. M. Crampin, G. E. Prince and G. Thompson, A geometric version of the Helmholtz conditions in time dependent Lagrangian dynamics, J. Phys. A: Math. Gen. 17 (1984) 1437–1447. Crossref, Google Scholar
5. P. Delanoe, On Bianchi identities, Rend. Circ. Mat. Palermo 51(2) (2002) 237–248. Crossref, Google Scholar
6. M. Jerie and G. E. Prince, Jacobi fields and linear connections for arbitrary second order ODE’s, J. Geom. Phys. 43 (2002) 351–370. Crossref, Web of Science, Google Scholar
7. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I (Wiley-Interscience, New York, 1963). Google Scholar
8. O. Krupková and G. E. Prince, Second order ordinary differential equation in jet bundles and the inverse problem of the calculus of variation, in Handbook of Global Analysis, eds. D. Krupka and D. Saunders (Elsevier, 2008). Crossref, Google Scholar
9. I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry (Springer-Verlag, 1993). Crossref, Google Scholar
10. E. Massa and E. Pagani, Jet bundle geometry, dynamical connections, and the inverse problem of Lagrangian mechanics, Ann. Inst. H. Poincaré, Phys. Theoret. 61 (1994) 17–62. Web of Science, Google Scholar
11. P. Renteln, Manifolds, Tensors, and Forms (Cambridge University Press, 2014). Google Scholar
12. J. Szilasi, R. Lovas and D. Kertész, Connections, Sprays and Finsler Structures (World Scientific, 2014). Abstract, Google Scholar
13. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. II, 2nd edn. (Publish or Perish, 1979). Google Scholar
14. S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989) 349–379. Crossref, Web of Science, Google Scholar
15. K. Yano and M. Kon, Structures on Manifolds (World Scientific, 1984). Link, Google Scholar