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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
##### Keywords:
equifocal submanifold; Lie group; symmetric space
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