Global differential invariants of nondegenerate hypersurfaces


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SAĞIROĞLU Y., GÖZÜTOK U.

TURKISH JOURNAL OF MATHEMATICS, vol.46, no.6, pp.2208-2230, 2022 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 46 Issue: 6
  • Publication Date: 2022
  • Doi Number: 10.55730/1300-0098.3264
  • Journal Name: TURKISH JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.2208-2230
  • Keywords: Hypersurface, Bonnet?s theorem, differential invariant
  • Karadeniz Technical University Affiliated: Yes

Abstract

Let {gij(x)}ni,j =1 and {Lij(x)}ni,j =1 be the sets of all coefficients of the first and second fundamental forms of a hypersurface x in Rn+1. For a connected open subset U C Rn and a C???-mapping x : U + Rn+1 the hypersurface x is said to be d-nondegenerate, where d E {1, 2, ... n}, if Ldd(x) =?? 0 for all u E U. Let M(n) = {F : Rn ???+ Rn | Fx = gx + b, g E O(n), b E Rn}, where O(n) is the group of all real orthogonal n x n-matrices, and SM(n) = {F E M(n) | g E SO(n)}, where SO(n) = {g E O(n) | det(g) = 1}. In the present paper, it is proved that the set {gij(x), Ldj(x), i, j = 1,2,. .., n} is a complete system of a SM(n + 1)-invariants of a d-non-degenerate hypersurface in Rn+1. A similar result has obtained for the group M(n + 1).