Matrix Differential Equations for Pseudo-orthogonal Groups

  • V. I. ChilinEmail author
  • K. K. Muminov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)


We consider a system of matrix differential equations whose nondegenerate solutions are O(npR)-equivalent, where O(npR) is the pseudo-orthogonal group of invertible linear transformations of \(R^n\). We show that the class of first columns of the set of matrices that are nondegenerate solutions of this system coincides with the class of O(npR)-equivalent paths in \(R^n\).


Pseudo-orthogonal group Regular path Equivalence of paths Matrix differential equation 


  1. 1.
    Aripov, R.G., Khadzhiev, J.: A complete system of global differential and integral invariants of a curve in Euclidean geometry, Izvestiya Vuzov. Russian Math. 7(542), 3–16 (2007)zbMATHGoogle Scholar
  2. 2.
    Chilin, V.I., Muminov, K.K.: The complete system of differential invariants of a curve in pseudo-Euclidean space. Dyn. syst. 3(31), 1–2, 135–149 (2013)Google Scholar
  3. 3.
    Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1990)Google Scholar
  4. 4.
    Khadzhiev, J.: Application of the Theory of Invariants to the Differential Geometry of Curves. FAN Publications, Tashkent (1988). (Russian)Google Scholar
  5. 5.
    Muminov, K.K.: Equivalence of paths with respect to the action of the symplectic group, Izvestiya Vuzov. Russian Math. 7(537), 27–38 (2002)Google Scholar
  6. 6.
    Muminov, K.K.: Equivalence of curves with respect to an action of the symplectic group, Izvestiya Vuzov. Russian Math. 6, 31–36 (2009)Google Scholar
  7. 7.
    Muminov, K.K., Chilin, V.I.: Equivalence of Curves in Finite-Dimensional Vector Spaces. LAP LAMBEPT Academic Publishing, Germany (2015)Google Scholar
  8. 8.
    Pontryagin, L.S.: Ordinary Differential Equations. London Addison-Wesley, Reading (1962)Google Scholar

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Authors and Affiliations

  1. 1.National University of UzbekistanTashkentUzbekistan

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