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Veronese curves and webs: interpolation. (English) Zbl 1128.53013

For a manifold \(V\) of dimension \(pc\) a familiy \(\{{\mathfrak F}_t\}_{t \in {\mathcal V}}\) of foliations of dimension \(pc-c\) parametrized by a Veronese curve \(\mathcal V\) is called a Veronese web. This notion was introduced by I. M. Gelfand and I. Zakharevich [J. Funct. Anal. 99, 150–178 (1991; Zbl 0739.58021)] for the case \(c=1\). Later F.-J. Turiel in a series of papers [see C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 10, 891–894 (1999; Zbl 0929.53015)] and A. Panasyuk [see Rep. Math. Phys. 50, No. 3, 291–297 (2002; Zbl 1042.53008)] developed this theory for arbitrary values of \(c\). Veronese webs give rise to ordinary \(k\)-webs [see M. A. Akivis and V. V. Goldberg, Differential geometry of webs. In: F. J. E. Dillen (ed.) et al. Handbook of differential geometry. Amsterdam: North-Holland (2000; Zbl 0968.53001)] if one restricts the parameter \(t\) to a finite set \(\{t_1,\cdots,t_k\}\), which can be considered as finite interpolations of a Veronese web. The authors discuss this connection in detail. They give a short proof of a theorem from Panasyuk’s paper which answers a conjecture posed by I. Zakharevich [“Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs”, arxiv:math/0006001] stating that the distribution \(\{{\mathfrak F}_t\}_t\) is integrable if and only if there are \(p+2\) values for which \(\{{\mathfrak F}_t\}_t\) is integrable.

MSC:

53A60 Differential geometry of webs
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References:

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