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Naturally reductive Riemannian homogeneous structures on some classes of generic submanifolds in complex space forms. (English) Zbl 0860.53032

Let \(M^n(c)\) be a non-flat complex space form. Geodesic hyperspheres, horospheres and tubes about totally geodesic \(M^n(c)\) are equipped with a naturally reductive homogeneous structure and hence, are naturally reductive homogeneous spaces in the simply connected case.
In the present paper, the author provides new examples of submanifolds of \(M^n(c)\) equipped with such a structure which are not (locally) symmetric. He proceeds as follows: First, let \(c>0\). Any Riemannian product of a finite number of odd-dimensional spheres can be embedded in an odd-dimensional sphere of an appropriate radius. This projects via the Hopf map onto a generic submanifold of a complex projective space. In this case the author gives the explicit expression of a naturally reductive structure on these submanifolds. Further, similar examples are constructed for the case \(c<0\) by using the corresponding Hopf map \(\pi:H^{2n+1}_1(r)\to\mathbb{C} H_n\) where \(H_1^{2n+1}(r)\) denotes the anti-de Sitter space of radius \(r\), and by considering products of the form \(S^{2m_k+1}(r_k)\times\dots \times S^{2m_1+1}(r_1)\times H^{2m_0+1}_1(r_0)\).

MSC:

53C40 Global submanifolds
53C30 Differential geometry of homogeneous manifolds
53B20 Local Riemannian geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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