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The geodesic complexity of a Riemannian manifold is a numerical isometry invariant that is determined by the structure of its cut loci. In this article we study decompositions of cut loci over whose components the tangent cut loci ﬁber in a convenient way. We establish a new upper bound for geodesic complexity in terms of such decompositions. As an application, we obtain estimates for the geodesic complexity of certain classes of homogeneous manifolds. In particular, we compute the geodesic complexity of complex and quaternionic projective spaces with their standard symmetric metrics. Keywords Geodesic complexity · Cut locus · Topological complexity · Motion planning Mathematics Subject Classiﬁcation 55M30 · 53C22 Contents 1 Introduction ............................................... 2 Geodesic complexity and the total cut locus ............................... 3 Fibered decompositions of cut loci ................................... 4 The total cut loci of symmetric spaces .................................. 5 Three-dimensional lens spaces ...................................... References .................................................. Stephan Mescher stephan.mescher@mathematik.uni-halle.de Maximilian Stegemeyer maximilian.stegemeyer@mis.mpg.de Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5, 06120 Halle (Saale), Germany Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany 123 S. Mescher, M. Stegemeyer 1 Introduction The geodesic complexity of a complete Riemannian manifold is an integer-valued isometry invariant. It is given as a geometric analogue of the notion of topological complexity as introduced by M. Farber in Farber (2003). Geodesic complexity was originally deﬁned by D. Recio-Mitter in the more general framework of metric spaces in Recio-Mitter (2021). Given a complete Riemannian manifold (M , g) we denote its space of length-minimizing geodesic segments by GM, seen as a subspace of the path space C ([0, 1], M ) with the compact-open topology. Consider the endpoint evaluation map ev : GM → M × M , ev(γ ) = (γ (0), γ (1)). The geodesic complexity of (M , g), denoted by GC(M , g), is deﬁned as the smallest integer k for which there exists a decomposition of M × M into locally compact subsets A ,..., A with each A admitting a continuous local section of ev. 1 k i In the same way that topological complexity is motivated by a topological abstrac- tion of the motion planning problem from robotics, geodesic complexity is motivated by an abstract notion of efﬁcient motion planning. Sections of ev can be seen as geodesic motion planners since they assign to a pair of points (p, q) ∈ M × M a length-minimizing path connecting these two points. As noted by Recio-Mitter in (Recio-Mitter 2021, p. 144), the main problem in determining the geodesic complexity of (M , g) lies in understanding the geodesic motion planning problem on its total cut locus. The latter is deﬁned as Cut(M ) ={(p, q) ∈ M × M | q ∈ Cut (M )}⊆ M × M , where Cut (M ) ⊂ M denotes the cut locus of p ∈ M with respect to the given metric g. In this article we introduce the notion of a ﬁbered decomposition of the total cut locus. If such a decomposition exists, it gives rise to a new upper bound for the geodesic complexity of M. The main applications of this upper bound are estimates for the geodesic complexity of certain homogeneous Riemannian manifolds. Similar situations were already studied by the authors in Mescher and Stegemeyer (2021). The upper bounds in the present article are however independent of the ones given in Mescher and Stegemeyer (2021). Various estimates from that article can be improved using the new results. Let (M , g) be a complete Riemannian manifold and consider the extended expo- nential map Exp : TM → M × M , Exp(v) = (pr(v), exp (v)), pr(v) where pr : TM → M denotes the bundle projection. We say that the total cut locus admits a ﬁbered decomposition if there is a decomposition of Cut(M ) into locally compact subsets A ,..., A , such that the restriction 1 k π := Exp | : A → A , i i i 123 Geodesic complexity via ﬁbered... −1 where A = Exp (A )∩Cut(M ), is a ﬁbration for each i ∈{1,..., k}. Here, Cut(M ) i i denotes the total tangent cut locus which will be deﬁned below. We will establish the following result. Theorem (Theorem 3.2) Let (M , g) be a complete Riemannian manifold. If the total cut locus Cut(M ) admits a ﬁbered decomposition A ,..., A with ﬁbrations 1 k π : A → A for i ∈{1,..., k}, then the geodesic complexity of M can be estimated i i i by GC(M , g) ≤ secat(π : A → A ) + 1. i i i i =1 Here, secat denotes the sectional category of a ﬁbration, which was introduced by A. Schwarz in Albert (1966) as the genus of a ﬁber space. Evidently, this raises the question whether there are any interesting cases of Rieman- nian manifolds whose total cut loci admit ﬁbered decompositions. For a homogeneous Riemannian manifold we will establish a tangible criterion on the cut locus of a single point implying that its total cut locus admits a ﬁbered decomposition. We will fur- ther show that each irreducible compact simply connected symmetric space satisﬁes this condition, providing a large class of examples whose total cut loci admit ﬁbered decompositions. By applying this result, we are able to compute the geodesic com- plexity of complex and quaternionic projective spaces with respect to their standard symmetric metrics. n n Theorem (Theorem 4.6) Let M = CP or HP equipped with its standard or Fubini- Study metric g , where n ∈ N. Then its geodesic complexity satisﬁes sym GC(M , g ) = 2n + 1. sym In particular, the geodesic complexity of (M , g ) equals the topological complexity sym of M. Moreover, using results by V. Ozols from Ozols (1974), we study the total cut locus of three-dimensional lens spaces with metrics of constant sectional curvature. We show that lens spaces of the form L(p; 1), where p ≥ 3, are further examples of homogeneous manifolds whose total cut loci admit ﬁbered decompositions. As these spaces are not globally symmetric, this shows that ﬁbered decompositions are not exclusively obtained in the globally symmetric case. A detailed analysis of the ﬁbrations involved in the ﬁbered decompositions of Cut(L(p; 1)) shows that 6 ≤ GC(L(p; 1), g) ≤ 7, see Theorem 5.8, where g is a metric of constant sectional curvature. This manuscript is organized as follows. In Sect. 2 we review the deﬁnitions of the total cut locus and of geodesic complexity and note some basic properties of these objects. 123 S. Mescher, M. Stegemeyer The central notion of a ﬁbered decomposition of the total cut locus is introduced in Sect. 3. In that section we also prove the above mentioned upper bound on geodesic complexity and study a criterion for the existence of a ﬁbered decomposition. Symmetric spaces are studied in Sect. 4. After recalling some properties of root systems and related notions we prove that the total cut loci of irreducible compact simply connected symmetric spaces admit ﬁbered decompositions and derive an upper bound on geodesic complexity. This will be applied to the examples of complex and quaternionic projective spaces and a particular complex Grassmannian. Finally, in Sect. 5 we discuss the total cut loci of three-dimensional lens spaces and study a ﬁbered decomposition to derive an upper bound on the geodesic complexity of these spaces. 2 Geodesic complexity and the total cut locus In this section we quickly introduce the basic notions of geodesic complexity and of the total cut locus. For more properties of geodesic complexity and of the relation between cut loci and geodesic complexity we refer to Recio-Mitter (2021) and Mescher and Stegemeyer (2021). Under a locally compact decomposition of a topological space X we understand a cover A ,..., A of X such that the A are pairwise disjoint and each A , i ∈{1,... k}, 1 k i i is a locally compact subspace of X. As usual we equip the path space C (I , M ) with the compact-open topology, where I =[0, 1] is the unit interval. For a Riemannian manifold (M , g) we let GM ⊆ C (I , M ) be the space of length-minimizing paths in M equipped with the subspace topology of C (I , M ), i.e. GM ={γ ∈ C (I , M ) | γ is a length-minimizing geodesic in M }. Deﬁnition 2.1 Let (M , g) be a complete Riemannian manifold and let ev : GM → M × M , ev(γ ) = (γ (0), γ (1)). (1) A local section of ev is called a geodesic motion planner. (2) Let B ⊆ M × M be a subset. The subspace geodesic complexity of B in (M , g) is deﬁned to be the smallest integer k for which there is a locally compact decompo- sition A ,..., A of B with the following property: for each i ∈{1,..., k} there 1 k exists a continuous geodesic motion planner A → GM. The subspace geodesic complexity of B in (M , g) is denoted by GC (B). If no such k exists, we put (M ,g) GC (B) := +∞. (M ,g) (3) The geodesic complexity of (M , g) is deﬁned to be the subspace geodesic complex- ity of M ×M itself and is denoted by GC(M , g), i.e. GC(M , g) = GC (M ×M ). (M ,g) Remark 2.2 (1) By the deﬁnition of topological complexity via locally compact decompositions, see (Farber 2008, Section 4.3), it is clear that the geodesic com- 123 Geodesic complexity via ﬁbered... plexity of a Riemannian manifold (M , g) is bounded from below by the topological complexity TC(M ) of M. (2) If the metric under consideration is apparent, then we will drop the metric from the notation and simply write GC(M ) := GC(M , g) or GC (B) := GC (B). M (M ,g) (3) Geodesic complexity was introduced by D. Recio-Mitter in Recio-Mitter (2021) for more general geodesic spaces, i.e. metric spaces in which any two points are connected by a length-minimizing path. Since every complete Riemannian manifold is a geodesic space, our deﬁnition is nothing but a particular case of Recio-Mitter’s deﬁnition. Note however that our deﬁnition of geodesic complex- ity differs from the one in Recio-Mitter (2021) by one. More precisely, while in Recio-Mitter (2021) a geodesic space of geodesic complexity k ∈ N is decom- posed into at least k + 1 locally compact subsets admitting geodesic motion planners, our deﬁnition requires the existence of a decomposition into k subsets having this property. As pointed out in Recio-Mitter (2021) the geodesic complexity of a Riemannian man- ifold (M , g) crucially depends on the cut loci of M. We next recall the notion of the cut locus of a point as well as the total cut locus and the total tangent cut locus of a Riemannian manifold. The latter two notions were introduced in Recio-Mitter (2021). Deﬁnition 2.3 Let (M , g) be a complete Riemannian manifold and let p ∈ M. (1) Let γ :[0, ∞) → M be a unit-speed geodesic with γ(0) = p. We say that the cut time of γ is t (γ ) = sup{t > 0 | γ | is minimal}. cut [0,t ] In case that t (γ ) < ∞ we say that γ(t (γ )) is a cut point of p along γ and cut cut that t (γ )γ( ˙ 0) ∈ T M is a tangent cut point of p. cut p (2) The set of tangent cut points of p is called the tangent cut locus of p and is denoted by Cut (M ) ⊂ T M. The set of cut points of p is called the cut locus of p and is p p denoted by Cut (M ). (3) The total tangent cut locus of M is given by Cut(M ) = Cut (M ) ⊆ TM . p∈M The total cut locus of M is deﬁned as Cut(M ) = ({ p}× Cut (M )) ⊆ M × M . p∈M Remark 2.4 Let (M , g) be a complete Riemannian manifold. (1) By deﬁnition of the Riemannian exponential map exp : T M → M at p ∈ M we have exp (tv) = γ (t ) for all t > 0 and v ∈ T M , v p 123 S. Mescher, M. Stegemeyer where γ is the unique geodesic starting at p with γ( ˙ 0) = v. Consequently, exp maps the tangent cut locus Cut (M ) onto the cut locus Cut (M ). p p (2) We recall the deﬁnition of the global Riemannian exponential map, see e.g. (Lee 2018, p. 128), which is given by Exp : TM → M × M , Exp(v) = (pr(v), exp (v)). pr(v) Here, pr : TM → M denotes the bundle projection. It is clear from the deﬁnitions that Exp maps the total tangent cut locus Cut(M ) onto the total cut locus Cut(M ). Finally, we want to note how the total cut locus of a Riemannian manifold (M , g) can be used to study the geodesic complexity of M. As Recio-Mitter argues in (Recio- Mitter 2021, Theorem 3.3) there is a unique continuous geodesic motion planner on (M × M ) Cut(M ). By (Błaszczyk and Carrasquel-Vera 2018, Lemma 4.2) the latter is an open subset of M × M, from which one derives the estimate GC (Cut(M )) ≤ GC(M ) ≤ GC (Cut(M )) + 1. (2.1) (M ,g) (M ,g) Hence, in order to ﬁnd bounds on the geodesic complexity of a complete Riemannian manifold (M , g) one can study the subspace geodesic complexity of its total cut locus Cut(M ). 3 Fibered decompositions of cut loci In this section we introduce the notion of a ﬁbered decomposition of the total cut locus of a Riemannian manifold M and show that such a ﬁbered decomposition of Cut(M ) can be used to derive upper and lower bounds on the geodesic complexity of M. After that we give a condition on the cut locus of a point p ∈ M of a homo- geneous Riemannian manifold which implies that the total cut locus admits a ﬁbered decomposition. Deﬁnition 3.1 Let (M , g) be a complete Riemannian manifold. (1) A locally compact decomposition A ,..., A of Cut(M ) is called a ﬁbered decom- 1 k position of Cut(M ) if the following holds: for each i ∈{1,..., k} the restricted exponential map π = Exp | : A → A i i i −1 is a ﬁbration, where A = Exp (A ) ∩ Cut(M ). i i (2) Similarly, if p ∈ M, then a locally compact decomposition B ,..., B of 1 k Cut (M ) is called a ﬁbered decomposition of Cut (M ) if p p exp | : B → B i i −1 is a ﬁbration, where B = exp (B ) ∩ Cut (M ). i i p 123 Geodesic complexity via ﬁbered... Here, under a ﬁbration we always understand a Hurewicz ﬁbration in the sense of homotopy theory. Next we will discuss how ﬁbered decompositions of cut loci yield new lower and upper bounds for geodesic complexity. Theorem 3.2 Let (M , g) be a complete Riemannian manifold. If the total cut locus Cut(M ) admits a ﬁbered decomposition A ,..., A with ﬁbrations π : A → A 1 k i i i for i ∈{1,..., k} as in Deﬁnition 3.1.(1), then the geodesic complexity of M can be estimated by GC(M ) ≤ secat(π ) + 1. i =1 Proof We begin by showing that continuous local sections of π induce continuous geodesic motion planners. Let C ⊆ A be a locally compact subset of A and assume i i that s : C → A is a continuous section of the ﬁbration π . In particular, we have for i i (p, q) ∈ C that Exp(s(p, q)) = (p, q). We deﬁne σ : C → GM by σ((p, q))(t ) = pr (Exp(ts(p, q))) for t ∈[0, 1], where pr : M × M → M denotes the projection onto the second component. This is clearly a geodesic motion planner. In order to see that map σ is also continuous note that the map σ : C × I → M,((p, q), t ) → pr (Exp(ts(p, q))) is continuous since it is a composition of continuous maps. By a general property of the compact-open topology, the continuity of σ implies the continuity of the induced map σ : C → GM, see e.g. [Bredon 2013, Theorem VII.2.4]. For each i ∈{1,..., k} we put m := secat(π ). Then, see e.g. (Mescher and i i Stegemeyer 2021, Lemma 4.1), for each i there is a locally compact decomposition C ,..., C of A for which there is a continuous section of π on each C , i ,1 i ,m i i i , j j ∈{1,..., m }. Since the sets A ,..., A form a decomposition of Cut(M ),wesee i 1 k that the sets {C | i ∈{1,..., k}, j ∈{1,..., m }} i , j i are a decomposition of Cut(M ) with each C locally compact. By the ﬁrst part of the i , j proof we see that each C admits a continuous geodesic motion planner. This shows i , j that k k GC (Cut(M )) ≤ m = secat(π ). (M ,g) i i i =1 i =1 123 S. Mescher, M. Stegemeyer Combining this inequality with the inequality (2.1) completes the proof. In the subsequent sections we will see examples of upper bounds on geodesic complexity by virtue of Theorem 3.2. The next result however shows how a ﬁbered decomposition of the total cut locus Cut(M ) gives rise to a lower bound on GC (Cut(M )). Before we state the result, we recall the deﬁnition of the veloc- (M ,g) ity map, see [Mescher and Stegemeyer 2021, Deﬁnition 3.1], i.e. the map given by v : GM → TM,v(γ) =˙ γ(0). The velocity map is continuous by [Mescher and Stegemeyer 2021, Proposition 3.2]. Furthermore, we recall that the sectional category of a ﬁbration p : E → B is deﬁned by considering open covers U ,..., U of B such that each U , i ∈{1,..., k} admits 1 k i a continuous local section of p. The geodesic complexity of a complete Riemannian manifold M however is deﬁned via locally compact decompositions of M × M.In order to compare these two concepts in the following theorem, we employ the notion of generalized sectional category as introduced by J. M. García Calcines in [García- Calcines 2019, Deﬁnition 2.1]. Deﬁnition 3.3 Let p : E → B be a ﬁbration. The generalized sectional category secat (p) is deﬁned as the smallest integer k for which there exists a cover A ,..., A g 1 k of B such that each A , i ∈{1,..., k}, admits a continuous local section of p. Note that the sets A in the above deﬁnition can be arbitrary subsets of B. García- Calcines shows in [García-Calcines 2019, Theorem 2.7] that if p : E → B is a ﬁbration and if E and B are absolute neighborhood retracts, one has secat (p) = secat(p). Theorem 3.4 Let (M , g) be a complete Riemannian manifold. Assume that the total cut locus Cut(M ) admits a ﬁbered decomposition A ,..., A with ﬁbrations π : A → A 1 l i i i for i ∈{1,..., l}. Furthermore, assume that all A and A are absolute neighborhood i i retracts. Then GC (Cut(M )) ≥ max{secat(π ) | i ∈{1,..., l}}. (M ,g) i Proof Let m ∈ N be the maximum of {secat(π ) | i ∈{1,..., l}} and choose i ∈ i 0 {1,..., l} such that secat(π ) = m. Assume that the assertion of the theorem is false. Then there are a locally compact decomposition B ,..., B of Cut(M ) with k < m 1 k and continuous geodesic motion planners s : B → GM for j ∈{1,..., k}.For j j i ∈{1,..., k} set C = B ∩A . It is possible that there are i ∈{1,..., k} with C =∅. i i i i By reordering the B we can arrange that C ,..., C =∅ and C ,..., C =∅ for i 1 r r +1 k some 1 ≤ r ≤ k.The sets C ,..., C form a cover of A .For j ∈{1,..., r } we 1 r i deﬁne a map σ : C → A ,σ = v ◦ s | , j j i j j C 0 j 123 Geodesic complexity via ﬁbered... where v denotes the velocity map. It is clear that σ is continuous. We claim that it is a section of A . For any (p, q) ∈ C the path s (p, q) is a minimal geodesic. Thus, i j j there is w ∈ Cut (M ) with s (p, q)(t ) = exp (tw). By deﬁnition of the velocity map, we obtain σ (p, q) = (v ◦ s )(p, q) = w j j and by deﬁnition of A it is clear that w ∈ A . Consequently, i i 0 0 (π ◦ σ )(p, q) = (Exp | ◦ σ )(p, q) = (p, exp (w)) = (p, q), i j j 0 A p which shows that σ is a continuous section of π . Hence, we obtain j i secat (π ) ≤ r ≤ k < m. g i Since π : A → A is a ﬁbration with A and A being absolute neighborhood i i i i i 0 0 0 0 0 retracts, we derive from [García-Calcines 2019, Theorem 2.7] that secat(π ) = secat (π )< m, i g i 0 0 which is a contradiction. This completes the proof. Corollary 3.5 Let (M , g) be a complete Riemannian manifold. Assume that π = Exp | : Cut(M ) → Cut(M ) Cut(M ) is a ﬁbration and assume that Cut(M ) and Cut(M ) are absolute neighborhood retracts. Then GC (Cut(M )) = secat(π ) and secat(π ) ≤ GC(M , g) ≤ secat(π ) + 1. (M ,g) Proof It is clear by Theorem 3.4 that GC (Cut(M )) ≥ secat(π ). (M ,g) The reverse inequality follows from the proof of Theorem 3.2. The second asserted inequality follows from equation (2.1). In Sect. 4 we will show that the symmetric metrics on complex and quaternionic projective spaces are examples for which the conditions of Corollary 3.5 are satisﬁed. In the following we will derive a tangible criterion in order to ﬁnd ﬁbered decom- positions of the total cut locus. In the setting of homogeneous Riemannian manifolds we want to use a ﬁbered decomposition of the cut locus of a point to obtain a ﬁbered 123 S. Mescher, M. Stegemeyer decomposition of the total cut locus, whose ﬁbrations will in fact be ﬁber bundles. We will see applications of this idea in Sects. 4 and 5. Note that if a compact group of isometries acts transitively on a Riemannian man- ifold, then the manifold is necessarily complete. If K is a group of isometries of a Riemannian manifold which ﬁxes a point p ∈ M, then k · Cut (M ) = Cut (M ) for p p all k ∈ K . Deﬁnition 3.6 Let (M , g) be a Riemannian manifold and assume that G is a group of isometries acting transitively on M.Let p ∈ M be a point and let K ⊆ G be its isotropy group. Let B ,..., B be a locally compact decomposition of Cut (M ).We 1 m p say that the decomposition is isotropy-invariant if k · B = B for all i = 1,..., m i i and all k ∈ K . In the following let (M , g) be a Riemannian manifold and let G be a group of isometries of M acting transitively on M. We denote the group action by : G × M → M. We shall use the shorthand notation = (g, ·) : M → M as well as (g, p) = g · p for g ∈ G, p ∈ M. Our aim is to use the homogeneity of M to construct a ﬁbered decomposition of the total cut locus Cut(M ) out of a ﬁbered decomposition of the cut locus of one single point in M. In the following, we ﬁx a point p ∈ M and let B ,..., B be a decomposition of 1 k Cut (M ) which is both isotropy-invariant and a ﬁbered decomposition such that the associated ﬁbrations B → B are ﬁber bundles for i ∈{1,..., k}. i i Let K be the isotropy group of p and let pr : G → M = G/K denote the canonical projection. For i ∈{1,..., k} set A ={(q, r ) ∈ Cut(M ) | r ∈ (B ) for some g ∈ G with pr(g) = q} i g i and A ={(q,v) ∈ Cut(M ) | v ∈ (D ) (B ) for some g ∈ G with pr(g) = q}. i g p i We further consider the maps π : A → M,π (q, r ) = q, π : A → M , π (q,v) = q, i ∈{1,..., k}. i i i i i i Lemma 3.7 In the present setting the following holds for each i ∈{1,..., k}: 1. π : A → M is a ﬁber bundle with typical ﬁber B . i i i 2. π : A → M is a ﬁber bundle with typical ﬁber B . i i i Note that by ﬁber bundle, we mean a ﬁber bundle in the continuous category. We do not assume that the sets B carry any differentiable structure. Proof We want to show that both A and A are locally trivial. Fix an i ∈{1,..., k} i i and let π : A → M,π (q, r ) = q, i i i 123 Geodesic complexity via ﬁbered... be the projection on the ﬁrst factor. Let U ⊆ M be an open set on which there exists a continuous section s : U → G of pr. Deﬁne ϕ : A | → U × B by i i U i −1 ϕ (q, r ) = (q, s(q) · r ) for (q, r ) ∈ A | . i i U This is a well-deﬁned map since if (q, r ) ∈ A , then there is a b ∈ B such that i i r = g · b for some g ∈ G with g · p = q. Therefore, by the isotropy invariance of the decomposition B ,..., B , 1 k −1 −1 s(q) · r = (s(q) g) · b ∈ B −1 since s(q) g ∈ K . Evidently, ϕ is a homeomorphism. For each point (q, r ) ∈ A i i there is such an open neighborhood U of q admitting a continuous section s : U → G of pr. Thus, the above construction shows that A → M is a continuous ﬁber bundle. The proof for A is analogous. One deﬁnes local trivializations of the form ψ : A | → U × B , where U is an open subset of M admitting a continuous section i i U i s : U → G of pr, by ψ (q,v) = (q,(D ) v) for (q,v) ∈ A . −1 i q i s(q) As for ϕ one shows that ψ is well-deﬁned and a homeomorphism. i i Theorem 3.8 Let (M , g) be a Riemannian manifold and G be a group of isometries of M acting transitively on M. Fix a point p ∈ M. Let B ,..., B be a decomposition 1 k of Cut (M ) which is both isotropy-invariant and a ﬁbered decomposition such that the associated ﬁbrations B → B are ﬁber bundles. For i = 1,...,klet C be the i i i typical ﬁber of the bundle B → B . Deﬁne the sets A ⊆ Cut(M ) as above. Then the i i i decomposition of Cut(M ) into A ,..., A is a ﬁbered decomposition. More precisely, 1 k the restriction Exp | : A → A is a ﬁber bundle with typical ﬁber C . i i i Proof Fix i ∈{1,..., k} and let p ∈ M. As discussed in the proof of Lemma 3.7, we can ﬁnd an open neighborhood U ⊆ M of p and local trivializations ϕ : A | → i i U U × B and ψ : A | → U × B .If ϕ and ψ are given as in that proof, then the i i i U i i i inverse of ϕ is explicitly given by −1 −1 ϕ : U × B → A | ,ϕ (q, b) = (q, s(q) · b), i i U i i where s : U → G is a local section of pr : G → M. We claim that the diagram Exp | A | i U A | A | i U i U ψ ϕ i i U × B U × B i i (id ,exp ) 123 S. Mescher, M. Stegemeyer commutes. To see this, let (q,v) ∈ A | . Then i U −1 ψ (q,v) = (q,(D −1 ) v) = (q,(D ) v) ∈ U × B . i q s(q) i s(q) By naturality of the exponential map, see [Lee 2018, Proposition 5.20], it thus holds that −1 −1 −1 (ϕ ◦ (id , exp ) ◦ ψ )(q,v) = ϕ (q, exp ((D ) v)) U i s(q) i p i p p −1 = (q, s(q)s(q) · exp (v)) = (q, exp (v)) = Exp(q,v). By assumption the restriction exp | : B → B is a ﬁber bundle. Hence, by choosing i i −1 an open subset V ⊆ B such that B | is trivial and considering ϕ (U ×V ) we obtain i i V an open set in A over which the map Exp | : A → A is trivial. Since A is covered i i i i by such trivializations, this proves the claim. 4 The total cut loci of symmetric spaces In this section we turn to the study of cut loci in irreducible compact simply connected symmetric spaces and show that the total cut locus of these spaces always admits a ﬁbered decomposition. Furthermore, we derive a new upper bound for the geodesic complexity of symmetric spaces. Note that this section is related to (Mescher and Stegemeyer 2021, Sect. 3) where the authors proved an upper bound for irreducible compact simply connected symmetric spaces in terms of the sectional category of the isometry bundle Isom(M ) → M over a symmetric space M and certain subspace geodesic complexities. The upper bound in the current section is derived independently of this previous result. We brieﬂy recall the most important notions related to root systems of symmet- ric spaces. Let M = G/K be a symmetric space with (G, K ) being a Riemannian symmetric pair. Denote the canonical projection by π : G → G/K M. There is a decomposition g = k ⊕ m of the Lie algebra g of G such that m T M is a π(e) linear isometry. We set o = π(e), where e is the unit element of G. Consider the complexiﬁcation g of g and choose a Cartan subalgebra h ⊆ g .A root of g is an C C C element α ∈ h of the dual space of h for which there exists an X ∈ g {0} with [H , X]= α(H )X for all H ∈ h. If a is a maximal abelian subalgebra of m, then consider the restriction α| of a root of g . If this restriction is non-zero, we call it a root of the symmetric pair (G, K ). We choose and ﬁx a set of simple roots of the symmetric pair (G, K ) and denote it by π(G, K ). We further let δ denote its highest root. See [Helgason 1978, Section X.3] or (Bröcker and Dieck 1995, Section V.4) for details on these notions. Due to the compactness of G we can choose an Ad -invariant inner product ·, · on g and 123 Geodesic complexity via ﬁbered... identify the roots with vectors in a via this inner product. Then a Weyl chamber of π(G, K ) can be deﬁned as W := {X ∈ a | γ, X > 0 ∀γ ∈ π(G, K )}. Note that one can deﬁne the other Weyl chambers by choosing other systems of simple roots. The Weyl group W (G, K ) of the symmetric pair (G, K ) is generated by the reﬂections s on the hyperplanes {H ∈ a | α(H ) = 0}. It is a ﬁnite group and acts simply transitively on the set of Weyl chambers of (G, K ). T. Sakai has studied the cut loci of compact simply connected symmetric spaces in Sakai (1978), see also Sakai (1977) and Sakai (1978). We summarize the main results. If there are two or more simple roots of (G, K ), put D := { ⊂ π(G, K ) | =∅,δ ∈/ }. In case there is only one simple root γ , this is then also the highest root and we set D := {{γ }}. If there are two or more simple roots, we set S := X ∈ W | γ, X > 0 ∀γ ∈ , γ, X = 0 ∀γ ∈ π(G, K ) , 2 δ, X = 1 for each ∈ D. In case there is a single simple root γ , we deﬁne S := {X ∈ a | 2γ, X= 1}. {γ } As usual, we denote by exp : g → G the exponential map of G and deﬁne exp : m → M , exp := π ◦ exp | . This in fact agrees with the Riemannian exponential at the point o under the canonical identiﬁcation m = T M and is often denoted by Exp. In order not to confuse it with the global Riemannian exponential map used in Section 3, we denote it by exp. For ∈ D set : K × S → M , (k, X ) = exp(Ad (X )), and : K × S → m, (k, X ) = Ad (X ). 123 S. Mescher, M. Stegemeyer Furthermore, we deﬁne Z := {k ∈ K | exp(Ad (X )) = exp(X ) ∀X ∈ S } and K ={k ∈ K | Ad (X ) = X ∀X ∈ S }. Evidently, Z and K are closed subgroups of K with K ⊆ Z . Sakai shows in (Sakai 1978, Proposition 4.10), that if ∈ D, then the map induces a differentiable embedding : K /Z × S → M . Deﬁne C := im for each ∈ D. The cut locus of the point o = π(e) ∈ M is then given by Cut (M ) = C ∈D see (Sakai 1978, Theorem 5.3). Moreover, the set {C } forms a locally compact ∈D decomposition of Cut (M ). Lemma 4.1 The map induces a continuous embedding : K /K × S → m and for C := im( ) we have that −1 C = exp (C ) ∩ Cut (M ). Proof By deﬁnition of K it is clear that induces a continuous map : K /K × S → m. To prove that is an embedding, we closely follow the proof of (Sakai 1978, Proposition 4.10). For the injectivity of ,let k, k ∈ K and X , X ∈ S such that Ad X = Ad X. We need to show that [k ]=[k] in K /K and that X = X . k k Clearly, it holds that Ad X = X . −1 k k Therefore, by (Helgason 1978, Proposition VI.2.2) we know that there is an element s of the Weyl group W (G, K ) of the Riemannian pair (G, K ) such that sX = X.But since X and X are in the closure of the same Weyl chamber, they have to be equal, −1 see (Sakai 1978, p. 131). This also shows that k k ∈ K ,so [k ]=[k] in K /K . In order to show that is an embedding, let (k ) be a sequence in K and n n∈N (X ) be a sequence in S such that Ad (X ) → Ad X for n →∞, where n n∈N k n k k ∈ K and X ∈ S . We want to show that [k ]→[k] in K /K and X → X for n →∞. Assume that this does not hold. Then by compactness of K there are k ∈ K and Y ∈ W and there are subsequences (k ) and (X ) with k → k 0 n i ∈N n i ∈N n 0 i i i and X → Y for i →∞ with ([k ], Y ) = ([k], X ). By continuity of Ad we have n 0 123 Geodesic complexity via ﬁbered... Ad Y = Ad X so as argued above for the injectivity, we obtain X = Y and [k ]=[k] k k 0 −1 in K /K which gives a contradiction. This shows the sequential continuity of , thereby yielding that is an embedding. Finally, by (Sakai 1978, p.133) we have that C = im( ) ⊆ Cut (M ). Moreover, it is clear by construction that exp(C ) = C . In order to show that −1 (exp| ) (C ) ⊆ C Cut (M ) let X ∈ Cut (M ) such that exp(X ) ∈ C . Then there is k ∈ K with k · exp(X ) = exp(Ad (X )) = exp(Y ) for some Y ∈ S .Weset q = exp(Y ) and X = Ad X. Clearly, X ∈ Cut (M ) and k o −1 since X ∈ exp (q) we have by (Sakai 1978, Lemma 4.7) that there is an h ∈ Z −1 with X = Ad(h)(Y ). But this implies that X = Ad(k h)(Y ) which shows that X ∈ C . It is clear by construction that the decomposition {C } of Cut (M ) is isotropy- ∈D invariant. The next theorem shows that it is a ﬁbered decomposition of Cut (M ). Theorem 4.2 Let M = G/K be an irreducible compact simply connected symmetric space with (G, K ) being a Riemannian symmetric pair and let p ∈ M. Then the cut locus of p admits a decomposition which is both isotropy-invariant and a ﬁbered decomposition with the associated ﬁbrations being ﬁber bundles. Proof As we have already argued, the decomposition of Cut (M ) into the C , ∈ D, is a decomposition into locally compact subsets and is isotropy-invariant. Hence, it remains to show that it is a ﬁbered decomposition. Let ∈ D and consider the map χ : K /K × S → K /Z × S ,χ(kK , X ) = (kZ , X ). We derive from Lemma 4.1 that the diagram K /K × S K /Z × S exp| C C commutes where the vertical arrows are homeomorphisms. It is well-known, see e.g. (Steenrod 1951, Theorem I.7.4), that the canonical map K /K → K /Z is a ﬁber bundle with typical ﬁber Z /K . Consequently, the above commutative diagram shows that exp| : C → C is a ﬁber bundle with typical ﬁber Z /K . Since this holds for all ∈ D we have shown that the decomposition {C } is a ﬁbered ∈D decomposition with the associated ﬁbrations being ﬁber bundles. Combining Theorems 3.8 and 4.2 we obtain the following. 123 S. Mescher, M. Stegemeyer Corollary 4.3 Let M be an irreducible compact simply connected symmetric space. Then the total cut locus Cut(M ) admits a ﬁbered decomposition and the associated ﬁbrations are ﬁber bundles. For ∈ D let A ⊆ Cut(M ) and A ⊆ Cut(M ) be the subsets of the total cut locus and the total tangent cut locus, resp., induced by the C as described in Sect. 3.The set π(G, K ) consists of precisely r = rank M elements. For each i ∈{1, 2,..., r } we set D := { ∈ D | # = i } and A := A . i i ∈D Note that by (Sakai 1978, Lemma 5.2), we have for all i ∈{1,..., r } that C ∩ C =∅ for ∈ D , It is easy to see that the same relation then holds for the A , i.e. A ∩ A =∅ for ∈ D , . (4.1) Therefore, if we have a locally compact decomposition of all A , ∈ D , then we can combine geodesic motion planners in the following way. Theorem 4.4 Let M be an irreducible compact simply connected symmetric space of rank r. Then the geodesic complexity of M can be estimated by GC(M ) ≤ max{secat(Exp | : A → A ) | ∈ D }+ 1. i =1 Proof Let i ∈{1,..., r } and assume that for each ∈ D we have a locally compact decomposition B ,..., B of A such that for each j ∈{1,..., k } there is a ,1 ,k continuous geodesic motion planner s : B → GM.Let m = max{k | ∈ D } , j , j i and set B =∅ for k < j ≤ m .For l = 1,..., m put , j i i B = B ,l ∈D and deﬁne a geodesic motion planner s : B → GM by l l s (q, r ) = s (q, r ) if (q, r ) ∈ B . ,l ,l It follows from (4.1) that this deﬁnes a continuous geodesic motion planner on B . Since the sets B ,..., B form a decomposition of A , this shows that GC (A ) ≤ m . 1 m i M i i Arguing as in the proof of Theorem 3.2, one further shows that k ≤ secat(Exp | : A → A ) ∀ ∈ D , 123 Geodesic complexity via ﬁbered... which in turn yields m ≤ max{secat(Exp | ) | ∈ D } for each i ∈{1, 2,..., r }. i i Eventually, we derive that r r r GC (Cut(M )) ≤ GC (A ) ≤ m ≤ max{secat(Exp | ) | ∈ D }. M M i i i i =1 i =1 i =1 Throughout the following, we shall always write = to indicate that two manifolds are diffeomorphic. We further let S denote the n-sphere with its standard differentiable structure for each n ∈ N. Example 4.5 Consider the complex Grassmannian Gr (C ) which is an irreducible compact symmetric space of rank 2. As shown in (Sakai 1978, p.143) and (Mescher and Stegemeyer 2021, Example 8.5), the cut locus Cut (M ) can be decomposed into 2 2 ∼ ∼ C = S × S , C = {∗} 1 2 and a six-dimensional manifold C . As discussed in (Mescher and Stegemeyer 2021, Example 8.5), these three spaces are simply connected. Note that D ={ }.The 1 1 2 decomposition of the cut locus of o induces a decomposition of Cut(M ) as in Sect. 3 and we shall call the induced sets A , A and A . In order to apply Theorem 4.4, 0 1 2 we need to ﬁnd upper bounds for secat(Exp | : A → A ) for i = 0, 1, 2. A i i Fix i ∈{0, 1, 2}. By (Albert 1966, Theorem 18), we have secat(π : E → B) ≤ cat(B) for any ﬁbration π where cat(B) is the Lusternik-Schnirelmann category of B. Consequently, we obtain secat(Exp | : A → A ) ≤ cat(A ). A i i i Note that Gr (C ) and C are simply connected, therefore A is simply connected i i since it is a ﬁber bundle over Gr (C ) with typical ﬁber C by Lemma 3.7. Therefore, we get the estimate dim(A ) dim(Gr (C )) + dim(C ) dim C i i i cat(A ) ≤ + 1 = + 1 = + 5 2 2 2 by (Cornea et al. 2003, Theorem 1.50). Explicitly, we obtain cat(A ) ≤ 8, cat(A ) ≤ 7 and cat(A ) ≤ 5. 0 1 2 Consequently, by Theorem 4.4, we see that GC(M ) ≤ cat(A ) + max{cat(A ), cat(A )}+ 1 = 16. 0 1 2 123 S. Mescher, M. Stegemeyer Note that this improves the upper bound in (Mescher and Stegemeyer 2021, Example 8.5). n n Theorem 4.6 Let M = CP or HP equipped with the standard or Fubini-Study metric, where n ∈ N. Then its geodesic complexity satisﬁes GC(M ) = 2n + 1. In particular, one has GC(M ) = TC(M ). n n Proof Since CP and HP are simply connected symmetric spaces of rank one, we know by (Sakai 1978, Theorem 5.3) and Corollary 4.3 that the restriction Exp | : Cut(M ) → Cut(M ) Cut(M ) is a ﬁbration. Moreover for n ≥ 2, the cut locus of a point satisﬁes n n−1 n n−1 ∼ ∼ Cut (CP ) CP and Cut (HP ) HP , = = p q n n where p ∈ CP and q ∈ HP , see (Arthur 1978, Proposition 3.35). By Lemma 3.7 n n n−1 n we see that Cut(CP ) is a ﬁber bundle over CP with typical ﬁber CP . Since CP is simply connected for each n ≥ 1, it follows that Cut(CP ) is simply connected as well for all n ≥ 2. By (Albert 1966, Theorem 18) and (Cornea et al. 2003, Theorem 1.50) we obtain dim(Cut(CP )) n n secat(Cut(CP ) → Cut(CP )) ≤ + 1 = 2n. Consequently by Theorem 3.2 we obtain n n n GC(CP ) ≤ secat(Cut(CP ) → Cut(CP )) + 1 ≤ 2n + 1 for n ≥ 2. Since TC(CP ) = 2n + 1 by (Farber 2006, Lemma 28.1), we obtain n n GC(CP ) = TC(CP ) = 2n + 1 n n for n ≥ 2. The argument for HP is similar, using that HP is 3-connected for all n ≥ 1 and that TC(HP ) = 2n + 1 by (Basabe et al. 2014, Corollary 3.16). 1 2 1 4 Finally, for n = 1 we have that CP is isometric to S and HP is isometric to S , 2 4 where both S and S are equipped with the standard metric. It is well-known that 2 4 GC(S ) = GC(S ) = 3, see (Recio-Mitter 2021, Proposition 4.1), so this proves the assertion for n = 1. 123 Geodesic complexity via ﬁbered... 5 Three-dimensional lens spaces In this section we show that the total cut locus of a lens space of the form L(p; 1) with a metric of constant sectional curvature admits a ﬁbered decomposition. It is thus an example of a homogeneous Riemannian manifold which has this property, but which is not a globally symmetric space, see e.g. (Gilkey et al. 2015, p. 105). We will use the explicit ﬁbered decomposition to derive an upper bound for the geodesic complexity of three-dimensional lens spaces of type L(p; 1). We start by studying the cut locus of a point in the lens space L(p; 1), which was explicitly described by S. Anisov in Anisov (2006). However, we give a self-contained exposition in this section, since we will need a detailed description of the tangent cut locus and of the cut locus in this setting. We consider the 3-sphere as a subspace of C , i.e. 3 2 S ={(z , z ) ∈ C | z z + z z = 1}. 1 2 1 1 2 2 3 4 In the following we will also consider S as embedded in R under the standard 2 4 identiﬁcation C = R . The special unitary group SU (2) acts transitively on the 3- sphere. Furthermore, for arbitrary p ≥ 3, we have an action of Z on S denoted by 3 3 : Z × S → S , where Z is the cyclic group with p elements, given by p p 2πim 2πim p p (m,(z , z )) → (e z , e z ). (5.1) 1 2 1 2 It is easy to see that this action is properly discontinuous. If we equip S with the standard metric, then is an action by isometries. Consequently, we can equip the 3 3 quotient L(p; 1) = S /Z with a metric for which π : S → L(p; 1) becomes a Riemannian covering. We henceforth always consider L(p; 1) as equipped with such a metric. The space L(p; 1) is called a lens space. Furthermore, note that the metric on L(p; 1) constructed in this way is a metric of constant sectional curvature. By the Killing-Hopf theorem all metrics of constant sectional curvature on L(p; 1) arise in this way, see e.g. (Lee 2018, Theorem 12.4 and Corollary 12.5). Note that the action of Z on S commutes with the action of SU (2). Thus, SU (2) acts on L(p; 1) and in particular this action is transitive, since it is already transitive on S . In the following we ﬁx the point p = π(1, 0) ∈ L(p; 1). Its isotropy group under the SU (2)-action on L(p; 1) is easily seen to be 2πik e 0 K = k ∈{0,..., p − 1} Z . (5.2) 2πik p 0 e Note that for more general lens spaces of the form L(p; q) where p and q are coprime with q = 1, see e.g. (Hatcher 2002, Example 2.43), the isometry group does not act transitively in general. See Kalliongis and Miller (2002) for details on the isometry groups of lens spaces. 123 S. Mescher, M. Stegemeyer In order to describe the cut locus of a point p ∈ L(p; 1), let us ﬁrst consider the more general situation of a Riemannian covering π : M → M. The following exposition closely follows (Ozols 1974, Section 3). It is well known that geodesics in M are mapped to geodesics in M under the Riemanian covering map π. Assume that M M/ where is a ﬁnite group of isometries of M acting properly discontinuously. Let d : M × M → R denote the distance function induced by the metric on M. For any two distinct points q, r ∈ M we set H ={u ∈ M | d(q, u)< d(r , u)}. q,r We recall from (Ozols 1974, Deﬁnition 3.1) that = H ⊆ M q q,g·q g∈ {e} is called the normal fundamental domain of centered at q. The following result by V. Ozols establishes a connection between normal fundamental domains and cut loci. Theorem 5.1 [(Ozols 1974, Corollary 3.11)] Let π : M → M be a Riemannian covering, let q ∈ M and let ⊂ M be its normal fundamental domain. If its closure satisﬁes ∩ Cut (M ) =∅, then q q Cut (M ) = π(∂ ). π(q) q Hence, to understand the cut locus of the point π(q) ∈ M M/ we can study the boundary of the normal fundamental domain .Let inj(T M ) ⊆ T M be the q q q domain of injectivity of the exponential map in M and put −1 := (exp | ) ( ) ⊆ T M . q q q q inj(T M ) Assume in the following that ∩ Cut (M ) =∅. Then exp maps homeomor- q q q phically onto , since the restriction of exp to inj(T M ) is a homeomorphism onto q q its image. With K := inj(T M ) ∪ Cut (M ) the diagram π(q) π(q) Dπ exp exp q ≈ π(q) commutes and one checks that the maps Dπ | : → K and exp | : q q q q q q are homeomorphisms. In particular, we see that ∂ is homeomorphic to the tangent cut locus Cut (M ) and that the exponential map π(q) exp | : Cut (M ) → Cut (M ) π(q) π(q) π(q) Cut (M ) π(q) 123 Geodesic complexity via ﬁbered... can be understood by considering π | : ∂ → Cut (M ). q π(q) In the following we denote by ·, · the standard inner product on R . The next lemma is easily shown by means of elementary geometry. Thus, we omit its proof. 3 4 Lemma 5.2 Let q, r ∈ S be two distinct points. Let u = q − r ∈ R and let E be 4 3 the 3-plane of points in R orthogonal to u. Then H ={v ∈ S |v, u > 0}. q,r We consider , the normal fundamental domain of Z centered at q = q p 0 (1, 0, 0, 0) ∈ S .For k ∈{0,..., p − 1}, we deﬁne 2πk 2πk u = q − k · q = 1 − cos( ), − sin( ), 0, 0 . k 0 0 p p By Lemma 5.2, it is clear that ={r ∈ S |u , r > 0for k = 1,..., p − 1}. q k Its boundary is = r ∈ S ∃k ∈{1,..., p − 1} with u , r = 0, u , r ≥ 0 ∀k ∈{1,..., p − 1} {k} . q k 0 k For l ∈{1,..., p − 1} and numbers 1 ≤ i < i < ... < i ≤ p − 1, we deﬁne 1 2 l (l) D = r ∈ S u , r=· · ·=u , r= 0, u , r > 0 ∀ j ∈{1,..., p − 1} {i ,..., i } . i i j 1 l i ,...,i 1 l 1 l It is clear that (l) = D . 0 i ,...,i 1 l l∈{1,...,p−1} 1≤i <...<i ≤p−1 1 l (l) (1) (1) ( p−1) Lemma 5.3 All sets of the form D are empty except D , D and D . i ,...,i 1 p−1 1,...,p−1 1 l ( p−1) (1) (1) Consequently, ∂ is the disjoint union of D , D and D . 1 p−1 1,...,p−1 Proof It is easy to see that ( p−1) 3 1 D ={(0, 0, x , y) ∈ S | (x , y) ∈ S }. 1,...,p−1 ( p−1) Hence, D is non-empty. For l ∈{1,..., p − 1}, l = ,weset 1,...,p−1 2πl 1 − cos( ) σ = . 2πl sin( ) 123 S. Mescher, M. Stegemeyer It can be seen directly that (1) 3 (1) 3 D ={(a,σ a, x , y) ∈ S | a > 0} and D ={(a,σ a, x , y) ∈ S | a > 0}. 1 p−1 1 p−1 (1) Note that σ =−σ .Let m ∈{2,..., p − 2}. We claim that D =∅. Assume p−1 1 m (1) that there is a point r = (a, b, x , y) ∈ D . Then, r , u = 0 implies that b = σ a if m = . For arbitrary m ∈{2,..., p − 2}, we get from u + u , r > 0 that 1 p−1 2π 2 1 − cos( ) a > 0 (5.3) which implies that a > 0. In case that p is even and m = , it can easily be seen that (1) a = 0, yielding a contradiction to inequality (5.3). Thus, D =∅. Therefore, we assume throughout the rest of the proof that m = . We consider two separate cases, starting with 2 ≤ m < . In this case we have σ > 0, so we see that b > 0. We i ϕ 2 write r = ( ae , x + iy) as an element of C with a > 0. It is clear that we have 2πm 1 − cos( ) tan ϕ = > 0 2πm sin( ) and we can choose ϕ ∈ (0, ). Since the third and fourth component of u are trivial, we can use the Euclidean inner product on R to compute that 2π 1 − cos( ) a cos(ϕ) u , r= , 2π a sin(ϕ) − sin( ) cos(ϕ − ) = , a −2sin( ) sin(ϕ − ) p p π π =−2 a sin( ) sin(ϕ − ), p p where we rotated the vectors by an angle of − to get the second equality. Note that π π by our assumption we have u , r > 0 which implies sin( ) sin(ϕ − )< 0. Since p p π π π π ϕ< by assumption, we want to show that ϕ> . Then sin( ) sin(ϕ − )> 0, 2 p p p which is thus a contradiction. The inequality ϕ> is equivalent to showing that tan(ϕ) > tan( ), i.e. that 2πm π 1 − cos( ) sin( ) p p > . (5.4) 2πm cos( ) sin( ) 123 Geodesic complexity via ﬁbered... p πm π Note that since m < ,wehave < . Consequently, 2 p 2 πm 2 π 2 2 cos( ) < 2 cos( ) . p p By standard trigonometry πm 2 2πm 2 cos( ) = 1 + cos( ) p p and therefore 2πm 2 π 2 2πm 1 − cos( ) < 2 cos( ) (1 − cos( )). p p p One checks by direct computation that this is equivalent to 2πm 2 π 2 2πm 2 π 2 (sin( )) (sin( )) <(1 − cos( )) (cos( )) . p p p p Since all squared terms were positive before squaring, we see that this is equivalent to 2πm π 2πm π sin( ) sin( )<(1 − cos( )) cos( ) p p p p which clearly implies the inequality (5.4). We thus get the desired contradiction in the p p case 2 ≤ m < . The case < m ≤ p − 2 can be treated similarly. One can argue 2 2 (l) similarly that all sets of the form D with 2 ≤ l ≤ p − 2 are empty. i ,...,i 1 l ( p−1) ( p−1) To shorten our notation we write D for D . Set p = π(q ) ∈ L(p; 1) 0 0 1,...,p−1 and recall that −1 Dπ ◦ (exp | ) : ∂ → Cut (L(p; 1)) q q p 0 q 0 0 0 ∂ is a homeomorphism. Moreover, the diagram Dπ Cut (L(p; 1)) q p 0 0 exp exp q p Cut (L(p; 1)) q p 0 0 commutes. Here, we obviously consider the restrictions of the maps to the spaces occurring in the diagram, which we drop from the notation for the sake of readability. (1) We denote the images of the sets D by (1) −1 (1) C = (Dπ ◦ (exp | ) )(D ), for i ∈{1, p − 1} 0 q ∂ i 0 q i 123 S. Mescher, M. Stegemeyer and similarly ( p−1) −1 ( p−1) C = (Dπ ◦ (exp | ) )(D ). 0 q ∂ Proposition 5.4 Let p ∈ N with p ≥ 3 and consider the lens space L(p; 1) with a Riemannian metric of constant sectional curvature. Let π : S → L(p; 1) be the corresponding Riemannian covering and put p := π(1, 0, 0, 0). (1) (1) ( p−1) (1) The sets C , C and C form a locally compact decomposition of the 1 p−1 (1) (1) tangent cut locus Cut (L(p; 1)). Moreover, C and C are homeomorphic 1 p−1 ( p−1) 1 to open 2-disks and C is homeomorphic to S . (2) The cut locus Cut (L(p; 1)) admits a locally compact decomposition into (1) (1) (1) C = π(D ) = exp (C ), i ∈{1, p − 1} i 0 i and ( p−1) ( p−1) p−1 C = π(D ) = exp (C ). (1) (1) The map exp | (1) : C → C is a homeomorphism for i ∈{1, p − 1}. Under 0 C ( p−1) ( p−1) 1 suitable identiﬁcations of C and C with S ,the map ( p−1) ( p−1) exp | : C → C (p−1) p C 1 1 can be identiﬁed with the standard p-fold covering of S by S . Hence, (1) ( p−1) Cut (L(p; 1)) = C C is a ﬁbered decomposition of Cut (L(p; 1)) and the associated ﬁbrations are ﬁber bundles. Proof The ﬁrst part is apparent given the identiﬁcation ∂ ≈ Cut (L(p; 1)) and q p 0 0 (1) (1) ( p−1) the characterization of the sets D , D and D in the proof of Lemma 5.3. 1 p−1 For the second part, we note that (1) (1) (1, D ) = D . p−1 1 (1) (1) Consequently, D and D are identiﬁed under π. Furthermore, the restriction of 1 p−1 (1) π to D , i ∈{1, p − 1} is a homeomorphism onto its image since it is continuous, (1) injective and a local homeomorphism. The same properties therefore hold for C , (1) C and the map exp under the identiﬁcation ∂ = Cut (L(p; 1)). Recall that q p p 0 0 p−1 ( p−1) 3 1 D ={(0, z) ∈ S | z ∈ S }, 123 Geodesic complexity via ﬁbered... 1 3 thus it is obviously homeomorphic to S and the Z -action on S becomes the standard 1 1 1 Z -action on S under this identiﬁcation. Since the map S → S /Z is a p-fold p p covering, this proves the last claim. In the following, we want to show that the ﬁbered decomposition of Cut (L(p; 1)) is isotropy-invariant with respect to the transitive SU (2)-action to obtain a ﬁbered decomposition of the total cut locus of L(p; 1) from Theorem 3.8. Lemma 5.5 Let (M , g) be a Riemannian manifold and let q ∈ M be a point. Further- more, let m ≥ 2 be an integer. Assume that G is a group of isometries of M which ﬁxes q. Let S ⊆ Cut (M ) be the set of points r ∈ Cut (M ) such that there are precisely m q q m distinct minimal geodesics between q and r. Then S is invariant under G. Proof Let ϕ : G × M → M denote the G-action and let : G × GM → GM denote the induced pointwise G-action, given by (g,γ )(t ) = (γ )(t ) = ϕ(g,γ (t )), for t ∈[0, 1], g ∈ G,γ ∈ GM . Let r ∈ S and g ∈ G. Since r is a cut point, s = ϕ(g, r ) ∈ Cut (M ).Let γ ,...,γ m q 1 m be the m distinct minimal geodesics between q and r. Then (γ ), ..., (γ ) are g 1 g m distinct minimal geodesics between q and s. If there was a minimal geodesic σ between q and s which is distinct from all (γ ), i ∈{1,..., m}, then −1 (σ ) would be a g i minimal geodesic joining q and r distinct from γ ,...,γ . This contradicts r ∈ S , 1 m m hence such a σ does not exist and we derive that s ∈ S as well. This proves the claim. (1) ( p−1) Corollary 5.6 The ﬁbered decomposition of Cut (L(p; 1)) = C C con- structed in Proposition 5.4 is isotropy-invariant. (1) ( p−1) Proof By Proposition 5.4 we can characterize C and C as (1) C ={q ∈ Cut (L(p; 1)) | there are precisely two minimal geodesics joining p and q}, p 0 ( p−1) C ={q ∈ Cut (L(p; 1)) | there are precisely p minimal geodesics joining p and q}. p 0 Therefore the isotropy invariance is a direct consequence by Lemma 5.5. It follows from Theorem 3.8 and Corollary 5.6 that there is a decomposition (1) ( p−1) of Cut(L(p; 1)) into sets A and A which form a ﬁbered decomposition of Cut(L(p; 1)). We now want to study this decomposition in greater detail. Recall that we denote the isotropy group of the SU (2)-action on L(p; 1) by K and computed it in equation (5.2). In order to better distinguish the various group actions, let 3 3 : SU (2) × S → S and ϕ : SU (2) × L(p; 1) → L(p; 1) be the actions of SU (2) on S and on L(p; 1), respectively. Recall that we denoted the Z -action on S be , see equation (5.1). If A ∈ SU (2) we shall also write p A 3 3 for the diffeomorphism (A, ·) : S → S and similarly for the other actions. 123 S. Mescher, M. Stegemeyer The ﬁbered decomposition of Cut(L(p; 1)) is given as follows. For l ∈{1, p − 1}, we have (l) (l) A ={(q, r ) ∈ Cut(L(p; 1)) | r ∈ ϕ (C ), A ∈ SU (2) such that pr(A) = q}, where pr : SU (2) → L(p; 1) is the canonical projection. We denote the preimages of (1) ( p−1) (1) ( p−1) A and A in the total tangent cut locus by A and A . Explicitly, we have (1) (1) (1) A ={(q,v) ∈ Cut(L(p; 1)) | v ∈ (Dϕ ) (C ∪ C ), A ∈ SU (2) such that pr(A) = q} A p 1 p−1 (1) (1) By Proposition 5.4, Corollary 5.6 and Theorem 3.8 we obtain that A → A is ( p−1) ( p−1) a 2-fold covering and that A → A is a p-fold covering, where we allow coverings to be trivial, i.e. the total space of the covering might not be connected. (1) We want to show that A consists of two connected components which implies that (1) (1) A → A is a trivial covering. (1) Lemma 5.7 The set C ⊆ T L(p; 1) is isotropy-invariant with respect to the induced SU (2)-action in the tangent bundle T L(p; 1). More precisely if A ∈ K, (1) (1) (1) then (Dϕ ) (C ) = C . The same holds for C . A p 1 1 p−1 (1) Proof Let x ∈ C ⊆ T L(p; 1) and A ∈ K , i.e. there is a k ∈{0,..., p − 1} such 1 0 that 2πik e 0 A = . 2πik 0 e (1) It holds that ( ◦ )(q ) = q . We want to show that (Dϕ ) (x ) ∈ C . Consider −k A 0 0 A p the following diagram. A −k 3 3 3 S S S exp exp exp q (q ) q 0 A 0 0 (D ) (D ) −k A q (q ) 0 A 0 3 3 3 T S T S T S q (q ) q 0 A 0 0 Dπ Dπ Dπ q (q ) q 0 A 0 0 T L(p; 1) T L(p; 1) T L(p; 1) p p p 0 0 0 (Dϕ ) id A p The lower two squares commute by deﬁnition of ϕ and the fact that the induced action of on L(p; 1) is trivial. The upper two squares commute by the naturality of the exponential map. Note that all arrows in the lower two squares are isomorphisms. 123 Geodesic complexity via ﬁbered... If we restrict to Cut (L(p; 1)) and to ∂ , respectively, we obtain a commutative p q 0 0 diagram −k A q q 0 0 −1 −1 Dπ ◦exp Dπ ◦exp q q q q 0 0 0 0 (Dϕ ) A p Cut (L(p; 1)) Cut (L(p; 1)). p p 0 0 (1) −1 By the proof of Lemma 5.3 we can write y = (exp ◦(Dπ) )(x ) ∈ D as q q 0 0 1 y = ((1 + i σ )a, z) where a > 0, z ∈ C, and where σ was deﬁned in the proof of Lemma 5.3. Then it follows that 4πik ( ◦ )(y) = (1 + i σ )a, e z , −k A 1 (1) (1) which is again an element of D . Consequently, (Dϕ ) (x ) ∈ C . The argument A p 1 1 (1) for C is analogous. p−1 By the previous lemma, the sets (1) (1) A ={(q,v) ∈ Cut(L( p; 1)) | v ∈ (Dϕ ) (C ), A ∈ SU (2) such that pr(A) = q} A q 1 1 and (1) (1) A ={(q,v) ∈ Cut(L(p; 1)) | v ∈ (Dϕ ) (C ), A ∈ SU (2) such that pr(A) = q} A q p−1 p−1 (1) (1) (1) (1) (1) are well-deﬁned. Moreover, we clearly have A = A A . Since A → A 1 p−1 (1) (1) is a ﬁber bundle by Theorem 3.8,wenow seethat A → A is a trivial 2-fold covering. This implies that (1) (1) secat(A → A ) = 1. Theorem 5.8 Let p ∈ N with p ≥ 3 and consider the lens space L(p; 1) with a metric of constant sectional curvature. Then 6 ≤ GC(L(p; 1)) ≤ 7. Proof M. Farber and M. Grant have shown in (Farber and Grant 2008, Corollary 15) that the topological complexity of L(p; 1) is TC(L(p; 1)) = 6, which yields GC(L(p; 1)) ≥ TC(L(p; 1)) = 6, see (Recio-Mitter 2021, Remark 1.9). By Theorem 3.2 we have (1) (1) ( p−1) ( p−1) GC(L(p; 1)) ≤ secat(A → A ) + secat(A → A ) + 1. (5.5) 123 S. Mescher, M. Stegemeyer (1) (1) ( p−1) As argued above, we have secat(A → A ) = 1. Recall that A is a circle bundle over L(p; 1), therefore it is 4-dimensional and we get ( p−1) ( p−1) ( p−1) secat(A → A ) ≤ cat(A ) ≤ 5 by (Albert 1966, Theorem 18) and (Cornea et al. 2003, Theorem 1.50). Thus, the inequality (5.5) gives the upper bound GC(L(p; 1)) ≤ 7. Funding Open Access funding enabled and organized by Projekt DEAL. 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Journal of Applied and Computational Topology – Springer Journals
Published: Sep 1, 2023
Keywords: Geodesic complexity; Cut locus; Topological complexity; Motion planning; 55M30; 53C22
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