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Complex equifocal submanifolds and infinite-dimensional anti-Kählerian isoparametric submanifolds. (English) Zbl 1089.53037
Let \(G/K\) be a symmetric space, where \(G\) is a connected semi-simple Lie group and \(K\) its maximal compact subgroup. By \(G^c\) and \(K^c\) denote their complexifications. The author calls \(G^c/K^c\) the anti-Kählerian symmetric space associated with \(G/K\). He also considers the infinite-dimensional anti-Kählerian space \((V^c, \text{Re}\langle\;,\;\rangle^c)\) as a complexification of a pseudo-Hilbert space \((V,\langle\;,\;\rangle)\).
Furthermore, let \(M\) be an immersed submanifold in \(N= G/K\) with abelian normal bundle \(T^\perp M\). For a parallel unit vector field \(v\) normal to \(M\) denote by \(\gamma_{v_x}\) the maximal geodesic such that the velocity between \(\dot\gamma_{v_x}(0)\) is equal to \(v_x\), \(x\in M\). Assume that the number of distinct focal radii along \(\gamma_{v_x}\) is finite, not equal to zero and is independent on the choice of the point \(x\in M\). Let \(\{r_i(x)\}\) be the set of all local radii along \(\gamma_{v_x}\) as functions on \(x\in M\). They are called focal radius functions for \(v\). If these functions are constant on \(M\) for any \(v\) then \(M\) is called an equifocal submanifold.
In this paper the study of complete real analytic equifocal submanifolds is reduced to that of infinite-dimensional anti-Kählerian isoparametric submanifolds.

MSC:
53C40 Global submanifolds
53C35 Differential geometry of symmetric spaces
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