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Invariants of Immersions on n-Dimensional Affine Manifold

Yıl 2024, Cilt: 37 Sayı: 2, 924 - 937

Djavvat Khadjiev Gayrat Beshimov İdris Ören

https://doi.org/10.35378/gujs.1037048

Öz

Main results: The system of Christoffel symbols of the connection of an immersion ξ:J→R^n of an n-dimensional manifold J in the n-dimensional linear space R^n is a system of generators of the differential field of all Aff(n)-invariant differential rational functions of ξ, where Aff(n) is the group of all affine transformations of R^n. A similar result have obtained for the subgroup SAff(n) of ⁡Aff(n) generated by all unimodular linear transformations and parallel translations of R^n. Rigidity and uniqueness theorems for immersions ξ:J→R^n in geometries of groups Aff(n) and SAff(n) were obtained. These theorems are given in terms of the affine connection and the volume form of immersions.

Anahtar Kelimeler

Connection Riemannian curvature tensor Invariant

Destekleyen Kurum

The Scientific and Technological Research Council of Turkey The Ministry of Innovative Development of the Republic of Uzbekistan (MID Uzbekistan)

Teşekkür

This work is supported by The Scientific and Technological Research Council of Turkey (T\"{U}B{\.I}TAK) under Grant Number 119N613 and The Ministry of Innovative Development of the Republic of Uzbekistan (MID Uzbekistan) under Grant Number UT-OT-2020-2.

Kaynakça

  • 1. P. J. Olver, C. Qu, Y. Yang, Feature matching and heat flow in centro-affine geometry, SIGMA 16 (2020) 1-22.
  • 2. O. Halimi, D. Raviv, Y. Aflalo, R. Kimmel, Chapter 7 - Computable invariants for curves and surfaces,Editor(s): Ron Kimmel, Xue-Cheng Tai,Handbook of Numerical Analysis, Elsevier, Volume 20,2019,Pages 273-314,.
  • 3. D.Khadjiev,Complete systems of differential invariants of vector fields in a euclidean space,Turk.J. Math. 34 (2010) 543-559.
  • 4. D. Khadjiev, Application of Invariant Theory to the Differential Geometry of Curves, Fan Publ., Tashkent, 1988. [in Russian](Zbl 0702.53002)
  • 5. H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton Univ. Press, Princeton-New Jersey, 1946.
  • 6. D.Khadjiev,On invariants of immersions of an n-dimensional manifoldin an n-dimensional pseudo- euclidean space, J. Nonliear Math. Phys. Vol. 17, Supp 01 (2010) 49-70.
  • 7.D.Khadjiev,O ̋.Pekşen,The complete system of global integral an ddifferential invariants for equi- affine curves, Differ. Geom.Appl. 20 (2004) 167–175.
  • 8. W. M. Boothby, An Introduction to Differential Manifolds and Riemannian Geometry, Academic Press, New York, 1975.
  • 9. O.V.Manturov,A multiplicative integral,J.SovietMath.55(1991)2042-2076.
  • 10. Y.Aminov,The Geometry of Vector Fields,Amsterdam.GordanandBreachSciencesPubl.2000.
  • 11. F.Antoneli, P. H. Baptistelli, A. P. S. Dias, M. Manoel, Invariant theory and reversible-equivariant vector fields, J. Pure Appl. Algebra 213 (2009) 649-663.
  • 12. P. H. Baptistelli, M. G. Manoel, Some results on reversible-equivariant vector fields, Cadornos De Matema’tica 6(2005) 237-263.
  • 13. J. L. Barbosa, W. Ferreira, K. Tenenblat, Submanifolds of constant sectional curvature in pseudo- Riemannian manifolds, Ann. Global Anal. Geom. 14 (1996) 381-401.
  • 14. A. A. Borisenko, Isometric immersions of space forms into Riemannian and pseudo-Riemannian space of constant curvature, Russ. Math. Surv. 56 (3)(2001) 425-497.
  • 15. Q. Chen, D. F. Zuo, Y. Chen, Isometric immersions of Pseudo-Riemannian space forms, J. Geom. Phys. 52 (3) (2004) 241-262.
  • 16. P.G.Ciarlet,F.Larsonneur,Ontherecoveryofasurfacewithprescribedfirstandsecondfundamental forms, J. Math. Pures Appl. 81 (2002) 167-185.
  • 17. M. J. Doffou, R. L. Grossman, The symbolic computation of differential invariants of polynomial vector field systems using trees. Proceedings of the 1995 International Symposium of Symbolic and Algebraic Computation, A. H. M. Levelt, editor, ACM, (1995)26-31.
  • 18. M. Golubitsky, I. N. Stewart, D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Springer-Verlag, Applied Mathematical Sciences 69, vol.2, New York, 1985.
  • 19. Z. Kose, M. Toda , E. Aulisa, Solving Bonnet problems to construct families of surfaces, Balkan J. Geom. Appl. 16(2011) 70-80.
  • 20. A. M. Li, U. Simon, G. Zhao, Global Affine Differential Geometry of Hypersurfaces, Walter de Gruyter, Berlin-New York, 1993.
  • 21. C. Mardare, On the recovery of a manifold with prescribed metric tensor, Anal. Appl. (Singap.) 1(2003) 433-453.
  • 22. S.Mardare,OnisometricimmersionsofaRiemannianspacewithlittleregularity,Anal.Appl.(Sin- gap.) 2(2004) 193-226.
  • 23. K.Nomizu,T.Sasaki,AffineDifferentialGeometry,CambridgeUniversityPress,Cambridge,1994.
  • 24. P.A.Schirokow,A.P.Schirokow,AffineDifferentialgeometrie,Leipzig,Teubner1962.[Zbl.106.147; Russ. original Zbl. 85.367]
  • 25. K.S.Sibirskii,IntroductiontotheAlgebraicInvariantsofDifferentialEquations,ManchesterUniver- sity Press, New York, 1988.
  • 26. I.N.Vekua,FundamentalsofTensorAnalysisandTheoryofCovariants,Nauka,Moscow,1978.[in Russian]

Yıl 2024, Cilt: 37 Sayı: 2, 924 - 937

Djavvat Khadjiev Gayrat Beshimov İdris Ören

https://doi.org/10.35378/gujs.1037048

Öz

Kaynakça

  • 1. P. J. Olver, C. Qu, Y. Yang, Feature matching and heat flow in centro-affine geometry, SIGMA 16 (2020) 1-22.
  • 2. O. Halimi, D. Raviv, Y. Aflalo, R. Kimmel, Chapter 7 - Computable invariants for curves and surfaces,Editor(s): Ron Kimmel, Xue-Cheng Tai,Handbook of Numerical Analysis, Elsevier, Volume 20,2019,Pages 273-314,.
  • 3. D.Khadjiev,Complete systems of differential invariants of vector fields in a euclidean space,Turk.J. Math. 34 (2010) 543-559.
  • 4. D. Khadjiev, Application of Invariant Theory to the Differential Geometry of Curves, Fan Publ., Tashkent, 1988. [in Russian](Zbl 0702.53002)
  • 5. H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton Univ. Press, Princeton-New Jersey, 1946.
  • 6. D.Khadjiev,On invariants of immersions of an n-dimensional manifoldin an n-dimensional pseudo- euclidean space, J. Nonliear Math. Phys. Vol. 17, Supp 01 (2010) 49-70.
  • 7.D.Khadjiev,O ̋.Pekşen,The complete system of global integral an ddifferential invariants for equi- affine curves, Differ. Geom.Appl. 20 (2004) 167–175.
  • 8. W. M. Boothby, An Introduction to Differential Manifolds and Riemannian Geometry, Academic Press, New York, 1975.
  • 9. O.V.Manturov,A multiplicative integral,J.SovietMath.55(1991)2042-2076.
  • 10. Y.Aminov,The Geometry of Vector Fields,Amsterdam.GordanandBreachSciencesPubl.2000.
  • 11. F.Antoneli, P. H. Baptistelli, A. P. S. Dias, M. Manoel, Invariant theory and reversible-equivariant vector fields, J. Pure Appl. Algebra 213 (2009) 649-663.
  • 12. P. H. Baptistelli, M. G. Manoel, Some results on reversible-equivariant vector fields, Cadornos De Matema’tica 6(2005) 237-263.
  • 13. J. L. Barbosa, W. Ferreira, K. Tenenblat, Submanifolds of constant sectional curvature in pseudo- Riemannian manifolds, Ann. Global Anal. Geom. 14 (1996) 381-401.
  • 14. A. A. Borisenko, Isometric immersions of space forms into Riemannian and pseudo-Riemannian space of constant curvature, Russ. Math. Surv. 56 (3)(2001) 425-497.
  • 15. Q. Chen, D. F. Zuo, Y. Chen, Isometric immersions of Pseudo-Riemannian space forms, J. Geom. Phys. 52 (3) (2004) 241-262.
  • 16. P.G.Ciarlet,F.Larsonneur,Ontherecoveryofasurfacewithprescribedfirstandsecondfundamental forms, J. Math. Pures Appl. 81 (2002) 167-185.
  • 17. M. J. Doffou, R. L. Grossman, The symbolic computation of differential invariants of polynomial vector field systems using trees. Proceedings of the 1995 International Symposium of Symbolic and Algebraic Computation, A. H. M. Levelt, editor, ACM, (1995)26-31.
  • 18. M. Golubitsky, I. N. Stewart, D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Springer-Verlag, Applied Mathematical Sciences 69, vol.2, New York, 1985.
  • 19. Z. Kose, M. Toda , E. Aulisa, Solving Bonnet problems to construct families of surfaces, Balkan J. Geom. Appl. 16(2011) 70-80.
  • 20. A. M. Li, U. Simon, G. Zhao, Global Affine Differential Geometry of Hypersurfaces, Walter de Gruyter, Berlin-New York, 1993.
  • 21. C. Mardare, On the recovery of a manifold with prescribed metric tensor, Anal. Appl. (Singap.) 1(2003) 433-453.
  • 22. S.Mardare,OnisometricimmersionsofaRiemannianspacewithlittleregularity,Anal.Appl.(Sin- gap.) 2(2004) 193-226.
  • 23. K.Nomizu,T.Sasaki,AffineDifferentialGeometry,CambridgeUniversityPress,Cambridge,1994.
  • 24. P.A.Schirokow,A.P.Schirokow,AffineDifferentialgeometrie,Leipzig,Teubner1962.[Zbl.106.147; Russ. original Zbl. 85.367]
  • 25. K.S.Sibirskii,IntroductiontotheAlgebraicInvariantsofDifferentialEquations,ManchesterUniver- sity Press, New York, 1988.
  • 26. I.N.Vekua,FundamentalsofTensorAnalysisandTheoryofCovariants,Nauka,Moscow,1978.[in Russian]
Toplam 26 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Mathematics
Yazarlar

Djavvat Khadjiev 0000-0001-7056-5662

Gayrat Beshimov 0000-0002-5394-2179

İdris Ören 0000-0003-2716-3945

Erken Görünüm Tarihi 9 Aralık 2023
Yayımlanma Tarihi
Yayımlandığı Sayı Yıl 2024 Cilt: 37 Sayı: 2

Kaynak Göster

APA Khadjiev, D., Beshimov, G., & Ören, İ. (t.y.). Invariants of Immersions on n-Dimensional Affine Manifold. Gazi University Journal of Science, 37(2), 924-937. https://doi.org/10.35378/gujs.1037048
AMA Khadjiev D, Beshimov G, Ören İ. Invariants of Immersions on n-Dimensional Affine Manifold. Gazi University Journal of Science. 37(2):924-937. doi:10.35378/gujs.1037048
Chicago Khadjiev, Djavvat, Gayrat Beshimov, ve İdris Ören. “Invariants of Immersions on N-Dimensional Affine Manifold”. Gazi University Journal of Science 37, sy. 2 t.y.: 924-37. https://doi.org/10.35378/gujs.1037048.
EndNote Khadjiev D, Beshimov G, Ören İ Invariants of Immersions on n-Dimensional Affine Manifold. Gazi University Journal of Science 37 2 924–937.
IEEE D. Khadjiev, G. Beshimov, ve İ. Ören, “Invariants of Immersions on n-Dimensional Affine Manifold”, Gazi University Journal of Science, c. 37, sy. 2, ss. 924–937, doi: 10.35378/gujs.1037048.
ISNAD Khadjiev, Djavvat vd. “Invariants of Immersions on N-Dimensional Affine Manifold”. Gazi University Journal of Science 37/2 (t.y.), 924-937. https://doi.org/10.35378/gujs.1037048.
JAMA Khadjiev D, Beshimov G, Ören İ. Invariants of Immersions on n-Dimensional Affine Manifold. Gazi University Journal of Science.;37:924–937.
MLA Khadjiev, Djavvat vd. “Invariants of Immersions on N-Dimensional Affine Manifold”. Gazi University Journal of Science, c. 37, sy. 2, ss. 924-37, doi:10.35378/gujs.1037048.
Vancouver Khadjiev D, Beshimov G, Ören İ. Invariants of Immersions on n-Dimensional Affine Manifold. Gazi University Journal of Science. 37(2):924-37.