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‘Spindles’ in symmetric spaces. (English) Zbl 1114.53053

The article deals with special families \((M^\xi_t)_{0\leq t\leq\delta}\) of submanifolds of a Riemannian symmetric space \(M= G/K\) \((G= \text{Isom}(M))\) of compact type. If \({\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}\) is the corresponding Cartan decomposition, \(K_0\) the identity component of \(K\) and \(p:= eK\), then \(\xi\) is chosen as a “canonical” vector of \({\mathfrak p}\cong T_pM\) and \(M^\xi_t\) is defined to be the orbit \(K_0\gamma(t)\), where \(\gamma:[0,\delta]\to M\) is supposed to be a simply closed geodesic with \(\dot\gamma(0)= \xi\). The submanifold \(M^\xi_t\) is called a knot, if \(M^\xi_t= \{\gamma(t)\}\) (obviously \(M^\xi_0\) and \(M^\xi_\delta\) are knots), and each subfamily \((M^\xi_t)_{a\leq t\leq b}\) between two consecutive knots is called a spindle in \(M\) associated with \(\xi\). The article is primarily concerned with the number of these spindles. Furthermore, if \(\xi\) satisfies \(\text{ad}(\xi)^3= -\text{ad}(\xi)\) and \(M\) is simply connected or the number of spindles is odd, then the submanifolds \(M^\xi_t\) are extrinsic symmetric, in particular they have parallel second fundamental form.

MSC:

53C40 Global submanifolds
53C35 Differential geometry of symmetric spaces
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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References:

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