Abstract
We use the method of equivariant moving frames to revisit the problem of normal forms and equivalence of nondegenerate real hypersurfaces \(M \subset {\mathbb {C}}^2\) under the pseudo-group action of holomorphic transformations. The moving frame recurrence formulae allows us to systematically and algorithmically recover the results of Chern and Moser for hypersurfaces that are either non-umbilic at a point \({{\varvec{p}}}\in M\) 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 the method of moving frames to construct new convergent normal forms for the intermediate but overlooked case of nondegenerate real hypersurfaces at singularly umbilic points, namely those umbilic points where the hypersurface is not identically umbilic in a neighborhood thereof.
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Notes
This is similar to the ambiguity in the curvature invariant \(\kappa \) of a curve C in the Euclidean plane \({\mathbb {R}}^2\), whose sign changes under a \(180^{\circ }\) rotation, and hence depends on the curve’s orientation.
For this purpose, we ignore the order 2 term \(z {\bar{z}}\).
The reader should not confuse our use of this term with the standard definition of generic submanifolds in CR geometry [12].
They would be required if one wishes to compute the explicit formulae for the normalized differential invariants through the equivariant moving frame calculus, but here we choose to work purely symbolically.
This describes what is known as a “coordinate cross-section”; more general cross-sections are rarely used and will not concern us here.
The multi-index J on \(v_J\) contains all lower case letters, in \(\{z,{\bar{z}},u\}\), whereas on \(V_J\) it denotes the multi-index with the corresponding capital letters, in \(\{Z,{\overline{Z}},U\}\), and should perhaps be denoted \(\iota (J)\). However, since the type of a multi-index subscript is always clear from context, we choose not to unnecessarily clutter the notation.
Note that when \(\ell = 0\), terms involving \(u^{\ell -1}\) are undefined, but have a vanishing coefficient and so do not appear.
A polynomial is superquadratic if it contains no constant or linear terms.
The non-symbolic moving frame calculus would write out the explicit expressions for these lifted invariants and explicitly solve (3.2) for the pseudo-group parameters, with the (far more complicated) process continuing in a similar fashion throughout.
The choice of 48 as the normalization constant is so that we can precisely reproduce the normal form (1.2) of Chern and Moser. Any other nonzero constant would work equally well, with corresponding modifications to the coefficients of the resulting formulae.
As always, we are ignoring contact forms and only writing out the horizontal components of the differential.
An important but difficult question is to determine a minimal set of generating differential invariants for a given (pseudo-) group action. If the set consists of a single differential invariant, it is obviously minimal. Otherwise, except in the case of curves (one-dimensional submanifolds), there is no known criterion for determining whether or not a given generating set is minimal.
This and subsequent formulae were obtained with the help of Mathematica. They are quite complicated and we have chosen not to write them down here. Details are available from the authors upon request.
As before, we suppress any contributions from contact forms.
In general, differentiated invariants can have many such formulae owing to the variety of syzygies among the differential invariants.
If more than one is nonzero, then one can construct several such expressions; their equivalence is a consequence of the various syzygies among \({\mathfrak {J}},{{\overline{{\mathfrak {J}}}}},{\mathfrak {K}}\) and their invariant derivatives.
One should keep in mind that the signature so constructed retains the sign ambiguities (4.5). These can be eliminated by replacing \({\mathfrak {J}}\) and \( V_{Z^j{\overline{Z}}{}^kU^\ell }\) for \(j+k\) odd by \({\mathfrak {J}}^2\) and \({\mathfrak {J}}\, V_{Z^j{\overline{Z}}{}^kU^\ell }\), respectively, when parametrizing the signature.
One should not confuse this definition of generic hypersurface M with the standard definition of generic manifold in CR geometry.
The transformation must be real so as not to complexify the translation generators.
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Acknowledgements
The authors would like to thank Howard Jacobowitz for his helpful comments and discussions during the preparation of this paper. We also thank Valerii Beloshapka for introducing us to [3] and helpful discussions. We further thank Niky Kamran for reading a draft version and sending many helpful comments. The research of the second author was supported in part by a grant from IPM, No. 1400510415.
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Olver, P.J., Sabzevari, M. & Valiquette, F. Normal Forms, Moving Frames, and Differential Invariants for Nondegenerate Hypersurfaces in \({\mathbb {C}}^2\). J Geom Anal 33, 192 (2023). https://doi.org/10.1007/s12220-023-01243-8
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DOI: https://doi.org/10.1007/s12220-023-01243-8