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Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics. (English) Zbl 1482.53092

On a Finsler manifold \((M,F)\), a geodesic \(c\) is called a closed geodesic if it is a closed curve \(c:S^1=\mathbb{R}/\mathbb{Z}\longrightarrow M\). The group \(S^1\) acts on \(c\) by \(\theta.c(t)=c(\theta+t)\), for any \(\theta\in S^1\). The \(m\)th iterate of a closed geodesic \(c\) is defined by \(c^m(t) = c(mt)\). If, for any \(m\geq2\), a closed geodesic \(c\) is not the \(m\)th iterate of any other closed geodesics then it is called a prime closed geodesic.
Suppose that \((M,F)=(S^n,F)\), for \(n>1\), is a Finsler sphere of constant flag curvature \(K=1\). The group \(\hat{G}=G\times S^1\) acts on the free loop space \(\Lambda M\) of \(M\), where \(G\) is the connected isometry group \(I_0(M,F)\). Suppose that all the prime closed geodesics of positive constant speed can be listed as the following finite set \[ \{\mathfrak{{B}}_i:=\hat{G}.\gamma_i| \ \ i=1,\cdots, n\}, \] which is called the Assumption \((F)\), in this paper. Suppose that \(H\), \(H_0\) and \(m\) denote the subgroup of \(I_0(M,F)\) preserving each closed geodesic, the identity component of \(H\), and the dimension of \(H\), respectively. The author proves that there exist at least \(m\) geometrically distinct orbits of the form \(\mathfrak{{B}}_i\) such that each union \(B_i\) of geodesics in \(\mathfrak{{B}}_i\) is a totally geodesic submanifold in \(M\) with a nontrivial \(H_0\)-action.

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C22 Geodesics in global differential geometry
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References:

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