# Variational Principles of General Connections with a Certain Deformation of Representations

- 50 Downloads

## Abstract

We investigate variational principles of general connections on principal bundles. In order to develop further a gauge theory by means of general connections, we introduce a certain deformation of representations of the structure group. This enables us to define exterior covariant derivatives on a space of sort of shifted fields. We construct action densities by using general connections, and deduce a sort of Lagrange’s equation and that of inhomogeneous field equation simultaneously. Due to the definition of the curvature of general connections, a new term will arise in the latter equation, which we demonstrate later to be an obstruction for current conservation law to hold. Finally, we explain that a theory of general connections is a natural means to describe so-called Higgs mechanism.

## Keywords

General connection variational principle Lagrange’s equation inhomogeneous field equation current conservation law Higgs mechanism## Mathematics Subject Classification

58E30 58E15 53C05 53C80## Notes

### Acknowledgements

I would like to express my gratitude to professor Akira Yoshioka for his valuable advices and suggestions.

## References

- 1.Abe, N.: General connections on vector bundles. Kodai Math. J.
**8**(3), 322–329 (1985)MathSciNetzbMATHCrossRefGoogle Scholar - 2.Abe, N.: Geometry of certain first order differential operators and its applications to general connections. Kodai Math. J.
**11**, 205–223 (1988)MathSciNetzbMATHCrossRefGoogle Scholar - 3.Bleecker, D.: Gauge Theory and Variational Principles. Dover Publications, Inc., Mineola (2005)zbMATHGoogle Scholar
- 4.Hamilton, M.J.D.: Mathematical Gauge Theory with Applications to the Standard Model of Particle Physics. Springer, Berlin (2018)Google Scholar
- 5.Kikuchi, K.: Regular general connections and Lorentzian twisted products. SUT J. Math.
**32**(1), 35–57 (1996)MathSciNetzbMATHGoogle Scholar - 6.Kitada, K.: General connections on principal bundles. JP J. Geom. Topol.
**20**(4), 333–367 (2017)Google Scholar - 7.Nadj, D.F., Moór, A.: On the connection of recurrence of metric, eigen tensors and transition of length in \(W\)–\(O_{n}\) spaces. Publ. de l’Inst. Math. [Elektronische Ressource]
**41**, 179–188 (1980)zbMATHGoogle Scholar - 8.Nagayama, H.: A theory of general relativity by general connections I. TRU Math.
**20**, 173–187 (1984)MathSciNetzbMATHGoogle Scholar - 9.Nagayama, H.: A theory of general relativity by general connections II. TRU Math.
**21**, 287–317 (1985)MathSciNetzbMATHGoogle Scholar - 10.Otsuki, T.: Tangent bundles of order 2 and general connections. Math. J. Okayama Univ.
**8**, 143–179 (1958)MathSciNetzbMATHGoogle Scholar - 11.Otsuki, T.: On general connections I. Math. J. Okayama Univ.
**9**, 99–164 (1960)MathSciNetzbMATHGoogle Scholar - 12.Otsuki, T.: A certain space-time metric and smooth general connections. Kodai Math. J.
**8**, 307–316 (1985)MathSciNetzbMATHCrossRefGoogle Scholar - 13.Otsuki, T.: Singular point set of a general connection and black holes. Math. J. Okayama Univ.
**30**, 199–211 (1988)MathSciNetzbMATHGoogle Scholar - 14.Otsuki, T.: Behaviour of geodesics around the singular set of a general connection. SUT J. Math.
**27**, 169–228 (1991)MathSciNetzbMATHGoogle Scholar - 15.Otsuki, T.: A family of Minkowski-type spaces with general connections. SUT J. Math.
**28**, 61–103 (1992)MathSciNetzbMATHGoogle Scholar - 16.Otsuki, T.: On a 4-space with certain general connection related with a Minkowski-type metric on \(R_{+}^{4}\). Math. J. Okayama Univ.
**40**, 187–199 (1998)MathSciNetzbMATHGoogle Scholar - 17.Yildirim, M., Bektas, M.: On Joachimsthal’s theorems in Riemann–Otsuki space \(R\)–\(O_{3}\). Turk. J. Math.
**38**, 753–763 (2014)zbMATHCrossRefGoogle Scholar