On the local stability of differential forms

Golubitsky, Martin; Tischler, David

doi:10.1090/S0002-9947-1976-0431243-X

On the local stability of differential forms
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by Martin Golubitsky and David Tischler

Trans. Amer. Math. Soc. 223 (1976), 205-221

DOI: https://doi.org/10.1090/S0002-9947-1976-0431243-X

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Abstract:

In this paper we determine which germs of differential s-forms on an n-manifold are stable (in the sense of Martinet). We show that when $s \ne 1$ or when $s = 1$ and $n \leqslant 4$ Martinet had found almost all of the possible examples. The most interesting result states that for certain generic singularities of 1-forms on 4-manifolds an infinite dimensional moduli space occurs in the classification of the 1-forms with this given singularity type up to equivalence by pull-back via a diffeomorphism.

References

M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518, DOI 10.1007/978-1-4615-7904-5
Martin Golubitsky and David Tischler, The nonexistence of globally stable forms, Proc. Amer. Math. Soc. 58 (1976), 296–300. MR 415656, DOI 10.1090/S0002-9939-1976-0415656-3

On the stability of differential forms

Jean Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 95–178 (French, with English summary). MR 286119, DOI 10.5802/aif.340
David G. Schaeffer, A regularity theorem for conservation laws, Advances in Math. 11 (1973), 368–386. MR 326178, DOI 10.1016/0001-8708(73)90018-2
Gordon Wassermann, Stability of unfoldings, Lecture Notes in Mathematics, Vol. 393, Springer-Verlag, Berlin-New York, 1974. MR 0410789, DOI 10.1007/BFb0061658

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