Xu, Ming Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics. (English) Zbl 1482.53092 Pac. J. Math. 302, No. 1, 353-370 (2019). 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. Reviewer: Hamid Reza Salimi Moghaddam (Isfahan) Cited in 1 Document MSC: 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics) 53C22 Geodesics in global differential geometry Keywords:Katok metric; Randers sphere; constant flag curvature; orbit of closed geodesics; totally geodesic submanifold; fixed point set PDFBibTeX XMLCite \textit{M. Xu}, Pac. J. Math. 302, No. 1, 353--370 (2019; Zbl 1482.53092) Full Text: DOI arXiv References: [1] ; Anosov, Proceedings of the International Congress of Mathematicians, II, 293 (1975) [2] 10.1007/s00208-009-0401-1 · Zbl 1187.53040 [3] 10.1007/978-1-4612-1268-3 [4] 10.4310/jdg/1098137838 · Zbl 1078.53073 [5] 10.1007/s10455-018-9641-1 · Zbl 1414.53025 [6] 10.1090/conm/196/02427 [7] 10.1007/s000290050009 · Zbl 0897.53052 [8] ; Bryant, Houston J. Math., 28, 221 (2002) [9] 10.1515/forum-2012-0032 · Zbl 1301.53050 [10] 10.1016/j.jde.2016.02.025 · Zbl 1405.53060 [11] 10.1007/s00526-016-1075-7 · Zbl 1381.53071 [12] ; Katok, Izv. Akad. Nauk SSSR Ser. Mat., 37, 539 (1973) [13] 10.1016/j.aim.2009.03.007 · Zbl 1172.53027 [14] 10.4310/jdg/1214442633 · Zbl 0658.53042 [15] 10.1007/s00208-003-0485-y · Zbl 1050.53063 [16] 10.1090/conm/196/02433 [17] 10.1016/j.aim.2012.04.006 · Zbl 1261.53038 [18] 10.4310/jdg/1424880983 · Zbl 1326.53110 [19] 10.1016/j.difgeo.2018.07.002 · Zbl 1404.53094 [20] 10.1016/j.geomphys.2017.10.005 · Zbl 1388.53036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.