On some flat connection associated with locally symmetric surface

Maria Robaszewska

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica (2014)

  • Volume: 13, Issue: 1, page 19-43
  • ISSN: 2300-133X

Abstract

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For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P(M,G). Associated with - satisfying some particular conditions - local basis of TM local connection form of such a connection is an R(G)-valued 1-form build from the dual basis ω1, ω2 and from the local connection form ω of ▽. The structural equations of (M,∇) are equivalent to the condition dΩ-Ω∧Ω=0. This work was intended as an attempt to describe in a unified way the construction of similar 1-forms known for constant Gauss curvature surfaces, in particular of that given by R. Sasaki for pseudospherical surfaces.

How to cite

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Maria Robaszewska. "On some flat connection associated with locally symmetric surface." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 13.1 (2014): 19-43. <http://eudml.org/doc/268837>.

@article{MariaRobaszewska2014,
abstract = {For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P(M,G). Associated with - satisfying some particular conditions - local basis of TM local connection form of such a connection is an R(G)-valued 1-form build from the dual basis ω1, ω2 and from the local connection form ω of ▽. The structural equations of (M,∇) are equivalent to the condition dΩ-Ω∧Ω=0. This work was intended as an attempt to describe in a unified way the construction of similar 1-forms known for constant Gauss curvature surfaces, in particular of that given by R. Sasaki for pseudospherical surfaces.},
author = {Maria Robaszewska},
journal = {Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica},
keywords = {principal fibre bundle; Gauss curvature; local connection form},
language = {eng},
number = {1},
pages = {19-43},
title = {On some flat connection associated with locally symmetric surface},
url = {http://eudml.org/doc/268837},
volume = {13},
year = {2014},
}

TY - JOUR
AU - Maria Robaszewska
TI - On some flat connection associated with locally symmetric surface
JO - Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
PY - 2014
VL - 13
IS - 1
SP - 19
EP - 43
AB - For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P(M,G). Associated with - satisfying some particular conditions - local basis of TM local connection form of such a connection is an R(G)-valued 1-form build from the dual basis ω1, ω2 and from the local connection form ω of ▽. The structural equations of (M,∇) are equivalent to the condition dΩ-Ω∧Ω=0. This work was intended as an attempt to describe in a unified way the construction of similar 1-forms known for constant Gauss curvature surfaces, in particular of that given by R. Sasaki for pseudospherical surfaces.
LA - eng
KW - principal fibre bundle; Gauss curvature; local connection form
UR - http://eudml.org/doc/268837
ER -

References

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  1. [1] J. Gancarzewicz, Zarys współczesnej geometrii rózniczkowej, Script, Warszawa 2010. Cited on 24. 
  2. [2] S. Kobayashi, K. Nomizu, Foundations of differential geometry, vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London 1963. Cited on 22. Zbl0119.37502
  3. [3] M. Marvan, On the spectral parameter problem, Acta Appl. Math. 109 (2010), no. 1, 239-255. Cited on 20. 
  4. [4] K. Nomizu, T. Sasaki, Affine differential geometry. Geometry of affine immersions. Cambridge Tracts in Mathematics 111, Cambridge University Press, Cambridge, 1994. Cited on 22. Zbl0834.53002
  5. [5] B. Opozda, Locally symmetric connections on surfaces, Results Math. 20 (1991), no. 3-4, 725-743. Cited on 21. Zbl0758.53005
  6. [6] B. Opozda, Some relations between Riemannian and affine geometry, Geom. Dedicata 47 (1993), no. 2, 225-236. Cited on 21 and 22. Zbl0781.53012
  7. [7] R. Sasaki, Soliton equations and pseudospherical surfaces, Nuclear Phys. B 154 (1979), no. 2, 343-357. Cited on 19 and 20. 
  8. [8] C.L. Terng, Geometric transformations and soliton equations, Handbook of geometric analysis, No. 2, 301-358, Adv. Lect. Math. (ALM), 13, Int. Press, Somerville, MA, 2010. Cited on 20. Zbl1213.37095
  9. [9] E. Wang, Tzitzéica transformation is a dressing action, J. Math. Phys. 47 (2006), no. 5, 053502, 13 pp. Cited on 20. Zbl1111.53002

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