“School of Mathematics”

Back to Papers Home
Back to Papers of School of Mathematics

Paper   IPM / M / 17216
School of Mathematics
  Title:   Normal forms, moving frames and differential invariants for nondegenerate hypersurfaces in $C^2$
  Author(s):  Masoud Masoud Sabzevari (Joint with P. J. Olver and F. Valiquette)
  Status:   Published
  Journal: J. Geom. Anal.
  Vol.:  33
  Year:  2023
  Pages:   1-46
  Supported by:  IPM
  Abstract:
We use the method of equivariant moving frames to revisit the problem of normal forms and equivalence of nondegenerate real hypersurfaces M�?�¢??C^2 under the pseudo-group action of holomorphic transformations. The moving frame recurrence formulae allow us to systematically and algorithmically recover the results of Chern and Moser for hypersurfaces that are either non-umbilic at a point p�?�¢??M or umbilic in an open neighborhood of it. In the former case, the coefficients of the normal form expansion, when expressed as functions of the jet of the hypersurface at the point, provide a complete system of functionally independent differential invariants that can be used to solve the equivalence problem. We prove that under a suitable genericity condition, the entire algebra of differential invariants for such hypersurfaces can be generated, through the operators of invariant differentiation, by a single real differential invariant of order 7. We then apply moving frames to construct new convergent normal forms for nondegenerate real hypersurfaces at singularly umbilic points, namely those umbilic points where the hypersurface is not identically umbilic around them.

Download TeX format
back to top
scroll left or right