Geodesic completeness of pseudo and holomorphic-Riemannian metrics on Lie groups

GEODESIC COMPLETENESS OF PSEUDO AND HOLOMORPHIC-RIEMANNIAN METRICS ON LIE GROUPS AHMED ELSHAFEI, ANA CRISTINA FERREIRA & HELENA REIS Abstract. This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler-Arnold formalism in the holomorphic setting. We study the real Lie group SL(2,R)and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group SL(2,C). 1. Introduction The study of left-invariant pseudo-Riemannian metrics on Lie groups and their quotients is an important topic in differential geometry with applications in cosmology and general relativity, for the case of Lorentzian signature. Even though the formalism regarding pseudo-Riemanannian geometry is very similar to the Riemannian one, a fundamental issue in this respect is that pseudo-Riemannian metrics often fail to be geodesically complete, even in the compact case. In mathematical physics, incompleteness might initially have been regarded as a failure of the model, however, singularities have become so common (consider, for instance, the classical Schwarzschild spacetime or the Penrose-Hawkings singularity theorems) that it is reasonable to expect incompleteness under general physical assumptions. In fact, the completeness or incompleteness of a metric becomes a fundamental property of the model (cf. [26] for a survey on geodesic completeness in the context of mechanical systems). The notion of geodesic flow on Lie groups can be traced back to Euler’s work on the motion of rigid bodies in R3with a fixed point. Euler showed that these motions can be described as geodesics on the special orthogonal group SO(3). Two centuries later, Arnold showed, in his seminal paper [1], that the motions of a rigid body with a fixed point can indeed be described as the geodesics of a left-invariant pseudo-Riemannian metric on a Lie group. Another important result to be noted is that, only a few years later, Marsden [18] proved that compact pseudo-Riemannian Lie groups (and more generally homogeneous spaces) are geodesically complete. The technique introduced by Arnold in [1] lends itself well to investigate the completeness of left-invariant pseudo-Riemannian metrics. More precisely, Arnold established that these geodesics are in one-to-one correspondence with integral curves of a certain quadratic (homogeneous) vector field on the corresponding Lie algebra (cf. Section 4). We will thus refer to this method as the “Euler-Arnold formalism” and to the quadratic vector field in question as the “Euler-Arnold vector field”. The Euler-Arnold formalism has a great advantage in the study of completeness of geodesics; the geodesics become described as the integrals curves of a quadratic (homogeneous) vector field on Rn. The study of the completeness of a vector field is played on the behavior at infinity. Since Rn (resp. Cn) has a simple compactification, the projective space RP(n)(resp. CP(n)), this allows us to have control on this behavior at infinity. Also, the fact that the vector field is algebraic (in fact, polynomial) is important. Indeed, there is a vast literature on algebraic differential equations that can be brought to bear. For example, certain ideas from [25] will be used in the course of this work. Date: February 8, 2023. 2020 Mathematics Subject Classification. Primary 53C22; Secondary 53C30, 53C50, 53C56, 37F75. Key words and phrases. geodesic (semi)completeness, Euler-Arnold equations, holomorphic metric. 1 2 A. ELSHAFEI, A.C. FERREIRA & H. REIS The task of classifying metrics on an arbitrary (non-compact) Lie group remains a daunting one, even in the semisimple case (see Section 9 for more details). Nevertheless, for the lowest dimensional simple Lie group SL(2,R)a complete classification can be indeed achieved. As a matter of fact, the problem of studying (in)complete metrics on SL(2,R)has a history that goes back a few decades. A first study of completeness of left-invariant pseudo-Riemannian metrics on SL(2,R)was developed by Guediri-Lafontaine [11] in 1995. Later, in 2008, Bromberg and Medina [4] provided the complete classification. A more geometric approach to the same problem was given by Tholozan [27] in 2014, as a part of his Ph.D. thesis. In this paper, we present an alternative approach to the problem of classification of geodesically complete left-invariant metrics on SL(2,R). This dynamical approach is build on ideas of [25] to estimate the size of the domains of definition of the geodesics, which we followed to obtain the conditions under which such a domain is strictly contained in R. This allowed us to reobtain the classification of complete left-invariant pseudo-Riemannian metrics previously provided in [4]. In fact, this allowed us to go further and also present a detailed study of completeness or incompleteness for every single geodesic of all the metrics in question. The structure of the paper can be essentially divided into three parts. The first part (Sections 2, 3, 4) is a discussion on (the not so well-known) holomorphic-Riemannian geometry of complex manifolds with special attention to Lie groups, leading up to the description of the Euler-Arnold formalism. The second part (Sections 5, 6, 7, 8) concerns the classification of completeness of left-invariant pseudo-Riemannian metrics on SL(2,R)and the dynamical behavior of their geodesic flows. The third and final part of the paper (Section 9) is concerned with left-invariant holomorphic metrics on SL(2,C). Sections 2 and 3 present the preliminary material necessary in the sequel in order to provide a self-contained exposition. Since the study of pseudo-Riemannian geometry and of real Lie groups is more familiar, we present our material in the holomorphic setting, bearing in mind that most of the constructions can find direct analogues in the real context and pointing out the differences where needed. In Section 4, we develop the Euler-Arnold formalism for completeness of holomorphic metrics. There are several approaches in the literature to establish this result; in our work, we used the methods (albeit in the real case) proposed by Tholozan [27] since they are more adapted to the complex setting. In the case of an orthogonal Lie group, the Euler-Arnold equations are more tractable and can be formulated in terms of a Lax-pair. The added feature that Lax equations (or, equivalently, that Euler-Arnold equations) come with two first integrals allowed us to establish the following general result. Theorem 1.1. Let Gbe a Lie group equipped with a bi-invariant pseudo-Riemannian metric. Then Gcan be endowed with a complete left-invariant pseudo-Riemannian metric of every possible signature. In Sections 5–8 we focus on SL(2,R). In Section 5, we present two important lemmas which allow us to control the orthogonality and bracket relations on sl(2,R). This outlines the special character of SL(2,R)and makes it possible to have an explicit description of the Euler-Arnold differential system. The study is conducted in four subcases corresponding to the normal forms of the isomorphism which relates the metric in question to the Killing form. Sections 6, 7 and 8 are devoted to the detailed study of geodesic (in)completeness for each of the subcases established in Section 5. For example, it was known that every metric whose associated isomorphism has a (unique real) eigenvalue with algebraic multiplicity 3 but geometric multiplicity 1admits an incomplete geodesic. We show in this paper that, for such a metric, there is no geodesic whose domain of definition is a finite interval ]a, b[for some a, b ∈Rwith a<b. To be more precise, we proved the following. Theorem 1.2. Consider a metric whose associated isomorphism has a (unique real) eigenvalue with algebraic multiplicity 3 but geometric multiplicity 1. Then there exists an invariant plane for the GEODESIC COMPLETENESS ON LIE GROUPS 3 geodesic flow over which all geodesics are complete. Furthermore, all the other geodesics are R+or R−complete, although not complete. This contrasts with the case where the isomorphism associated with the metric has two complex (non-real) eigenvalues. In this case the domain of definition of almost all geodesics is a finite interval, ]a, b[for some a, b ∈Rwith a<b. In fact, we proved the following. Theorem 1.3. Consider a metric whose associated isomorphism has two complex (non-real) eigenvalues. Then there exists an invariant plane for the geodesic flow over which all geodesics are complete. Furthermore, there exists an invariant plane over which some of the geodesics are R+ or R−complete and the others have a finite interval as domain of definition. All the remaining geodesics are incomplete with a finite interval as domain of definition. The final section, Section 9, is devoted to the study of SL(2,C). We start by presenting brief considerations why a direct and similar approach to that of SL(2,R)is unfeasible. We then turn our attention to the complex Lie group SL(2,C)and obtain a complete characterization for completeness of holomorphic-Riemannian left-invariant metrics. More precisely, we have the following theorem. Theorem 1.4. Let qbe the holomorphic metric on sl(2,C)defined by q(X, Y ) = B(ΦX, Y ), where Bis the Killing form and Φis a B-self-adjoint isomorphism. Then, qis a complete holomorphic metric if and only if Φhas an eigenvalue whose eigenspace has dimension at least 2. As a corollary, we show that every complex simple Lie group can be endowed with an incomplete holomorphic metric. As seen from the example of SL(2,C), complete holomorphic metrics are “rare” and, therefore, the problem of classifying semicomplete holomorphic metrics becomes more interesting. A vector field or, equivalently, a differential system, is said to be semicomplete if its solutions admit a maximal domain of definition in C(that need not coincide with C), see [22]. If a vector field is complete, then it is necessarily semicomplete. Although the converse does not hold, a semicomplete vector field can be completed in the sense that it can be realized by a complete vector field on a suitable space that needs not be a Lie group any more (see [21], [9], [12] for further details). In fact, this new space is not necessarily Hausdorff. It is perhaps relevant to understand these completions as they are important in the theory of transformation groups and may be relevant in applications to physics or other fields of geometry. Finally, as an unexpected outcome of our work concerning the general theory of semicomplete vector fields, we provide an example of a family with 2-complex parameters of iso-spectral quadratic semicomplete vector fields on C3, up to linear conjugation with the terminology of [12], see Theorem 9.3. Note that this family is rather different from the 2-parameter family provided in [13] and sits closer to the framework of “quadratic elliptic foliations on the complex projective plane” as in [10]. 2. Holomorphic-Riemannian geometry The foundations of pseudo-Riemannian geometry are well-known and there is extensive literature on the subject (see, for instance, [20]). In this section. we will briefly establish preliminary notions and results of its holomorphic counterpart, bearing in mind that most of definitions and results apply analogously in the smooth real setting. We will follow the approach of LeBrun [16] and point out the differences where relevant. The topic of holomorphic-Riemannian geometry has been studied in more recent years in the works of Biswas, Dumitrescu and Zeghib (cf. [3, 6, 7]). Let Mbe a complex manifold of (complex) dimension nwith holomorphic tangent and cotangent bundles denoted, respectively, by T0Mand T∗ 0M. A holomorphic-Riemannian metric (or, simply, holomorphic metric) on Mis a holomorphic covariant symmetric 2-tensor i.e. a holomorphic symmetric section g:M−→ T∗ 0M⊗T∗ 0Mwhich is also non-degenerate. 4 A. ELSHAFEI, A.C. FERREIRA & H. REIS In terms of local coordinates, a holomorphic metric gcan be expressed as follows. If Uis a domain of Msuch that we have a chart U −→ Cnwith coordinates (z1,··· , zn), then g|Ucan be written as g|U= n X i,j=1 gijdzi⊗dzj with ¯ ∂gij = 0 (i.e. gis holomorphic) and gij =gji (i.e. gis symmetric), for every i, j ∈ {1, . . . , n}. Clearly, such a tensor carries no signature, but it is still possible to define the notion of nondegenerate metric by prescribing that the map T0M−→ T∗ 0Mgiven by X7−→ g(X, −)is an isomorphism between the tangent and the cotangent holomorphic bundles. As in the real setting, this condition is translated by det[gij]= 0. A holomorphic-Riemannian manifold is thus a pair (M, g)where Mis a complex manifold and g is a holomorphic-Riemannian metric on M. Remark that the notion of holomorphic metric should not be confused with that of Hermitian metric. Observe also that the real part of a holomorphic metric is a pseudo-Riemannian metric of signature (n, n). Indeed, it is always possible to find a local basis of holomorphic vectors (X1,··· , Xn)such that g(Xi, Xj) = δij and then (X1,··· , Xn, iX1,··· , iXn)is a local basis for the underlying real manifold which is orthonormal for Re gwith signature (n, n). It is a well-known fact that a compact Lorenztian manifold has Euler characteristic equal to zero, and this is simply due to the fact that there must exist a nowhere vanishing (timelike) vector field. The existence of a holomorphic metric for compact complex manifolds is also restrictive in terms of its topology, cf. [6]. The existence of such a metric fixes an isomorphism between the holomorphic tangent and the holomorphic cotangent bundles, which implies that the canonical bundle K= Λn(T∗ 0M)of Mis isomorphic to its dual, the anticanonical bundle K∗, and this yields the vanishing of the first Chern class of M, since c1(M) = c1(K) = −c1(K∗). With the notion of holomorphic metric understood, we can now introduce the homolorphic LeviCivita connection. It can easily be proved, just like in the real setting that there exists a unique affine connection, which we will call the Levi-Civita connection, that is torsion-free and preserves the holomorphic metric, cf. [16]. Moreover, this connection is also defined by the Kozsul formula as in pseudo-Riemannian geometry. Given a holomorphic-Riemannian manifold (M, g), an isometry is defined to be a biholomorphic map ϕ:M−→ Mwhich preserves g, that is, ϕ∗g=g. If the biholomorphism ϕis defined only locally (between two open subsets of M) we say that ϕis a local isometry. As in the real case, the set of local isometries is a pseudo-group for composition. A similar proof to that of the classical result of pseudo-Riemannian geometry shows that the following three statements are equivalent for a holomorphic vector field X. (i) The (local) flow of Xpreserves g, i.e., it acts by isometries; (ii) ∇Xis a pointwise g-skew symmetric endomorphism of T0M; (iii) LXg= 0, where Lrepresents the Lie derivative. A vector field which satisfies any of the above conditions is called a Killing vector field. Note that the set of Killing vector fields forms a Lie algebra with respect to the standard Lie bracket of holomorphic vector fields. A notion of parallel transport can also be obtained in a classical fashion. Consider γ:A−→ M, A⊆C, an immersed (i.e. ˙γ(t)= 0, for all t∈A) holomorphic curve, and let ∇˙γdenote the covariant derivative along γ. Definition 2.1. The curve γis called a geodesic if ˙γis parallel along γ, that is, if ∇˙γ˙γ= 0. Using local coordinates and considering the Christoffel symbols Γk ij, the curve γis a geodesic of (M, g)if and only if it satisfies the equations ¨γk(t) + ˙γj(t)˙γi(t)Γk ij(γ(t)) = 0, which are perfectly analogous to the geodesic equations of pseudo-Riemannian geometry. GEODESIC COMPLETENESS ON LIE GROUPS 5 Using the theorem of existence and uniqueness of ordinary differential equations, we know that given a point pin Mand a vector vin the holomorphic tangent space of Mat p, there exists a ball B(0, δ)centered at 0∈Cwith radius δ > 0such that γ:B(0, δ)−→ Mis the unique geodesic verifying γ(0) = pand ˙γ(0) = v. We can now introduce the notion of completeness and semicompleteness for geodesics. Definition 2.2. Let (M, g)be a holomorphic-Riemannian manifold and ∇˙γ˙γ= 0 be the geodesic equation of (M, g). We say that the geodesic equation is semicomplete on an open set U⊆Mif for every p∈Uand v∈TpM, there exists a neighborhood Vpof 0∈Cand a curve γ:Vp−→ Uwhich satisfies the following conditions: (i) γ(0) = p,˙γ(0) = vand ∇˙γ˙γ= 0; (ii) for every sequence {ti}i∈N⊂Vp⊂Cwhich converges to a point ˆ tin the boundary of Vp, the sequence {γ(ti)}i∈Nescapes from every compact set of U. Definition 2.3. For a holomorphic-Riemannian manifold (M, g), the geodesic equation is said to be complete if there is a map Γ: C×M−→ Msuch that for every pin M, the curve γ(t) = Γ(t, p) satisfies condition (i) of Definition 2.2. The notion of geodesic completeness is clearly analogous to that of the real setting. Our definition of geodesic semicompleteness here is a particular case of the general definition for (germs of) vector fields on complex manifolds, which was introduced by Rebelo in [22]. In fact, Definition 2.2 can be rephrased by saying that the vector field on TM associated with the geodesic flow is semicomplete on TU ⊆TM. Note that for real ordinary differential equations the definition of semicompleteness is moot and the notion of maximal domain (or interval) of definition is well understood. This is an important difference between real and complex ODEs, since for a generic complex ODE the standard gluing procedure of the local domains of definition might lead to multivalued solutions (for more details, the reader is referred to the survey [24]). However, in the case where the geodesic equation is semicomplete on M, owing to condition (ii) of Definition 2.2, we will say that Vpis the maximal domain of definition of the geodesic γ, and our geodesics are well-defined in the sense of being univalued. The following lemma provides an interesting relation between Killing fields and geodesics; it will also be useful in Section 4. Lemma 2.4. Let (M, g)be a holomorphic-Riemannian manifold. If Xis a Killing field and γis a geodesic then g(Xγ(t),˙γ(t)) is constant. Proof. By definition of connection along a curve we have that d dt g(X, ˙γ) = g(∇˙γX, ˙γ) + g(X, ∇˙γ˙γ). Since γis a geodesic the above expression simplifies to d dt g(X, ˙γ) = g(∇˙γX, ˙γ).Since Xis a Killing field then d dt g(X, ˙γ) = −g(X, ∇˙γ˙γ)Thus, using again the fact that γis a geodesic, d dt g(X, ˙γ) = 0 and g(X, ˙γ)is constant. □ 3. Holomorphic-Riemannian Lie groups We will now focus on the case where our manifold is a Lie group. A very detailed exposition of this topic, in the (perhaps less familiar) complex setting can be found in the textbook by Lee, [17]. Here we will only recall the basic material needed for our purposes. As in the previous section, we will make our exposition in the complex case while bearing in mind the analogous definitions and results for the real case and pointing out the differences where needed. The definition of a complex Lie group is very similar to that of a real Lie group, but instead of considering smooth group operations, we must consider holomorphic ones. More precisely, a connected complex manifold Gwhich is also equipped with a group structure is said to be a complex Lie group if the multiplication and the inversion operations are holomorphic maps. 6 A. ELSHAFEI, A.C. FERREIRA & H. REIS Let Gbe a complex Lie group with identity e∈G. As usual, we will denote by Lg(resp. Rg, Cg) the left translation (resp. right translation, conjugation) map by g∈Gon G. Take Xto be a vector field on the Lie group G. Recall that Xis said to be left-invariant (resp. right-invariant) if it coincides with its pullback by left translations (resp. right translations). Naturally, given a vector x∈TeG, it can be extended to a left-invariant vector field Xby lefttranslating the tangent vector xto other points of G. This allows us then to identify the tangent space of Gat the identity, g=TeG, with the space of left-invariant vector fields on G. Also, it is simple to see that such an identification is a C-linear map, thus endowing gwith the structure of a complex Lie algebra. Let us recall the adjoint representation and the infinitesimal adjoint representation of a Lie group Gwhich we will denote, as usual, by Ad and ad, respectively. The adjoint representation of Gis the map Ad : G−→ GL(g)where Adgis the linear map satisfying Adg(x)=(DeCg)(x).Thus, the adjoint representation of Gis the derivative at the identity of the conjugation map Cg(viewed as an element of Aut(G), the automorphism group of G). Note that, with the standard complex structure on GL(g)≃GL(n, C), where n= dim G, it is possible to prove that Ad is a holomorphic map. The infinitesimal adjoint representation of G(or, equivalently, the adjoint representation of the Lie algebra gassociated to G) is the map ad : g−→ End(g)where adxis defined as adxy= (DeAd(x))(y).The infinitesimal adjoint representation of Gis then the derivative at the identity of the adjoint representation. Moreover, we have the remarkable fact that adxy= [X, Y ]e where x, y are vectors in TeGand X, Y are the left-invariant vector fields associated to xand yin the identification of g=TeGwith the Lie algebra of left-invariant vector fields. From henceforth, we will simply write, where convenient and as is standard, adxy= [x, y]. Remark 3.1. If Gis a matrix group, that is, a Lie subgroup of the general linear group GL(m, C), for some m∈N, then the adjoint representation is simply given by Adg(x) = gxg−1,for every g∈Gand x∈g; also the infinitesimal adjoint representation is simply the usual commutation of matrices, i.e., for every x, y ∈g,adx(y) = xy −yx. Consider now a holomorphic-Riemannian metric qon the complex Lie group G. The metric qis said to be left-invariant if all left translations are isometries, that is, if (Lg)∗q=q, for all g∈Gor, more concretely, if qh(x, y) = qgh ((DhLg)x, (DhLg)y), for all points g, h of Gand all vectors x, y of ThG. Analogously, a metric qis right-invariant if the right translations are isometries. It is easy to see that there is a one-to-one correspondence between left-invariant holomorphicRiemannian metrics on the group Gand non-degenerate complex bilinear forms on the corresponding Lie algebra g. Let us also recall the special case of bi-invariant metrics. A holomorphic-Riemannian metric is said to be bi-invariant if it is both left-invariant and right-invariant. Note that, for a connected group G, this is equivalent to having adxbe skew-symmetric with respect to q, that is, (1) q(adxy, z) + q(y, adxz)=0 for all x, y, z ∈g. The bilinear form κdefined by κ(x, y) = Tr(adx◦ady)for every x, y ∈g, where Tr stands for the trace of an endomorphism g−→ g, is called the Killing form of the Lie algebra g. The Killing form always satisfies the ad-invariance condition of Equation (1). Also, a famous result of Cartan tells us that Gis a semisimple Lie group if and only if its Killing form is nondegenerate. Therefore, for complex semisimple Lie groups we have a preferred bi-invariant holomorphicRiemannian metric. GEODESIC COMPLETENESS ON LIE GROUPS 7 Consider now any Lie group (real or complex, not necessarily semisimple) with Lie algebra g that can be equipped with a bi-invariant metric B. In the real setting, such groups are known as orthogonal, quadratic or quasi-classical Lie groups, [19]. We will now show that the set of leftinvariant metrics qon gis in one-to-one correspondence with B-self-adjoint isomorphisms Φ : g−→ g via the equality q(x, y) = B(Φx, y), for every x, y ∈g. Given a B-self-adjoint isomorphism Φ, it is clear that the equation above defines a non-degenerate symmetric bilinear form (possibly of different signature than that of B, in the real setting) on g. Conversely, given a metric qwe can define the isomorphism Φby the following commutative diagram g g∗ g g∗ Aq Φid AB where Aq(resp. AB) is the standard isomorphism between gand g∗given by q(resp. B). For a Lie group Gequipped with a left-invariant metric, the following lemma is an immediate consequence of the definitions but yet a noteworthy fact. Lemma 3.2. If Xis a right-invariant vector field then Xis a Killing field. Proof. Let Xbe a right-invariant vector field. We start by observing that if φ(t)is the one-parameter subgroup of Xe∈gthen the flow Ψt Xof Xis given by Ψt X(g) = φ(t)g=Lφ(t)g. By definition of Lie derivative, LXq= lim t→0 1 t(Ψt X)∗q−q But (Ψt X)∗q=L∗ φ(t)qfrom the above and L∗ φ(t)q=qsince the metric is left-invariant. The claim then follows. □ 4. The Euler-Arnold formalism In his celebrated article of 1966, [1], Arnold showed that the motions of a rigid body with fixed point can be seen as geodesics of a Lie group equipped with a left-invariant metric. This was a generalization of the result obtained by Euler for the particular case of rigid motions on R3. For this reason, the result of Theorem 4.1 (Equation (2) below) is known as the Euler-Arnold equation for geodesics of Lie groups. For the proofs, we will follow the approach of N. Tholozan in his Ph. D. thesis [27], which is very suitable to our complex setting. Let Abe an open domain in Cand γ:A−→ Gbe an immersed holomorphic curve in G. Using left translations, we can define the associated curve x:A−→ gin the Lie algebra gof G, for every t∈A, as follows x(t) = Dγ(t)Lγ−1(t)˙γ(t). Theorem 4.1 (Arnold, [1, 2]).Let (G, q)be a holomorphic-Riemannian Lie group. The curve γ:A−→ Gis a geodesic if and only if the associated curve x:A−→ gsatisfies, for every t∈A, the equation (2) ˙x(t) = ad† x(t)x(t), where ad† x(t)denotes the formal adjoint of adx(t)with respect to q. Proof. For simplicity, we will present the proof for matrix groups only. We recall that, by Ado’s theorem, every Lie algebra is isomorphic to the Lie algebra of a matrix group with the usual commutator, so we trust that the reader will not find our proof too restrictive. 8 A. ELSHAFEI, A.C. FERREIRA & H. REIS If Gis a matrix group and γ:A−→ Gis a curve in Gthen its associated curve x:A−→ gis simply given by the expression x(t) = γ−1(t) ˙γ(t). Suppose that γ:A−→ Gis a geodesic. Let zbe any element in the Lie algebra g. By Lemma 3.2, the right-invariant vector field Zγ(t):= zγ(t)is a Killing field along the curve γ. Also, according to Lemma 2.4, q(Zγ(t),˙γ(t)) is constant. Since the metric qis left-invariant, then c=qZγ(t),˙γ(t)=qγ−1(t)Zγ(t), γ−1(t) ˙γ(t)=qAdγ−1(t)z, x(t) for some c∈Cand for all t∈A. Taking the derivative with respect to t, we have that 0 = d dtqAdγ−1(t)z, x(t)=q˙x(t),Adγ−1(t)z+qx(t),d dtAdγ−1(t)z. From the equation above, by taking derivatives and noting that d dt γ−1(t) = −γ−1(t)˙γ(t)γ−1(t)and thus d dtAdγ−1(t)z=γ−1(t)z˙γ(t)−γ−1(t) ˙γ(t)γ−1(t)zγ(t),we get q˙x(t),Adγ−1(t)(z)=−qx(t), γ−1(t)z˙γ(t)−γ−1(t) ˙γ(t)γ−1(t)zγ(t). Now using the definitions of Ad and of x(t)we can rearrange our equation to q˙x(t),Adγ−1(t)z=−qx(t),Adγ−1(t)zx(t)−x(t)Adγ−1(t)z. Recalling that for matrix groups the adjoint map ad is given by the commutator, we obtain q˙x(t),Adγ−1(t)z=qx(t),adx(t)Adγ−1(t)z. Finally by taking the formal adjoint, we can write q˙x(t),Adγ−1(t)z=qad† x(t)x(t),Adγ−1(t)z. Now using the fact that zis a generic element in gand Adgis surjective for all g∈Gand also that qis non-degenerate, we can conclude that ˙x(t) = ad† x(t)x(t). Conversely, suppose that x(t)is an integral curve of the Euler-Arnold equation and take v=x(0). There exists a unique geodesic γin Gsuch that γ(0) = eand ˙γ(0) = v. Then, by the statement proved above, y(t) = γ−1(t)γ(t)is also an integral curve of the Euler-Arnold equation. Since y(0) = γ−1(0)˙γ(0) = v, by uniqueness, we conclude that x(t) = y(t), in an open neighborhood of 0∈C. Thus xis an associated curve in gof a geodesic of G.□ We remark that the theorem above, which we will call, as is usual in the literature, the EulerArnold theorem, is valid for any Lie group equipped with a left-invariant metric. For Lie groups which can be equipped with a bi-invariant metric, we have the following simplification of the EulerArnold equation. Proposition 4.2. Let Gbe an orthogonal Lie group equipped with the bi-invariant metric B. Let qbe a left-invariant metric on Gand Φbe the B-self-adjoint isomorphism defined by q(y, z) = B(Φy, z), for all y, z ∈g. A curve γ:A−→ Gis a geodesic in Gif and only if its associated curve x:A−→ g in gis an integral curve of Φ ˙x(t) = [Φx(t), x(t)]. Proof. Let zbe any element in g. Since x(t)satisfies the Euler-Arnold equation, we have that q( ˙x(t), z) = q(ad† x(t)x(t), z). It thus follows that B(Φ ˙x(t), z) = B(Φx(t),adx(t)z).Since Bis biinvariant we can write that B(Φ ˙x(t), z) = B([Φx(t), x(t)], z)and hence the claim follows. □ Recall that a first integral of a complex differential equation is a holomorphic function which is constant along its solutions. The Euler-Arnold equation comes with the following first integrals. Proposition 4.3. The functions I(x) = B(Φx, x)and J(x) = B(Φx, Φx)are first integrals of the Euler-Arnold equation Φ ˙x(t) = [Φx(t), x(t)]. GEODESIC COMPLETENESS ON LIE GROUPS 9 Proof. It suffices to prove that d dt I(x(t)) = 0 and d dt J(x(t)) = 0. We have that d dtI(x(t)) = d dtB(Φx(t), x(t)) = B(Φ ˙x(t), x(t)) + B(Φx(t),˙x(t)). Using the fact that Φis B-self-adjoint and also the Euler-Arnold equation, we get that d dtI(x(t)) = 2B(Φ ˙x(t), x(t)) = 2B([Φx(t), x(t)], x(t)) and since Bis bi-invariant then d dt I(x(t)) = 0. The result is analogous for J(x(t)).□ The following result is known as the Lax-pair formulation of the Euler-Arnold equation. Corollary 4.4. Let Gbe an orthogonal Lie group as formulated above. The geodesics of Gare in one-to-one correspondence with curves in gwhich satisfy the equation ˙z(t) = [z(t),Φ−1z(t)] . Furthermore, the functions F(z) = B(z, z)and G(z) = B(z, Φ−1z)are first integrals of the Lax-pair equation ˙z(t)=[z(t),Φ−1z(t)]. Proof. This is immediate from the two previous propositions by taking z= Φx.□ As can be seen from the discussion above, for orthogonal Lie groups with a pseudo or holomorphicRiemannian metric, the geodesic equation, which is for general manifolds an ODE of order 2, becomes an ODE of order 1 (or, equivalently, a vector field) in Euclidean space which comes with the added feature of having two first integrals. This allows us to prove the following result in the real setting. Theorem 4.5. Let Gbe a Lie group equipped with a bi-invariant pseudo-Riemannian metric B. Consider the set of left-invariant metrics for which the isomorphism Φis diagonalizable. Then, for every possible signature, there is an open set of eigenvalues of Φwhich corresponds to a set of complete left-invariant pseudo-Riemannian metrics. Proof. Suppose that Bhas signature (p, q)with p+q=n, where n= dim G. Fix, for the remainder of the proof, an orthogonal basis (e1,··· , ep, f1,··· , fq)such that B(ei, ei) = 1 = −B(fj, fj). Consider a pseudo-Riemannian metric qsuch that q(x, y) = B(Φx, y)where Φis a linear isomorphism which is represented by a diagonal matrix with respect to our fixed basis. Supposing that Φ−1= diag(ν1,··· , νp, µ1,··· , µq)then qis represented by diag(1/ν1,··· ,1/νp,−1/µ1,··· ,−1/µq). Notice that the signature of qis determined by the signs of νi, µj. If (z1,··· , zp, w1,··· , wq)stands for the chosen coordinates of z∈gthen the first integrals of Corollary 4.4 are given by I1(z) = z2 1+···+z2 p−w2 1−···−w2 q I2(z) = ν1z2 1+···+νpz2 p−µ1w2 1−··· −µqw2 q, and they are linearly independent for a generic metric. Suppose that qhas signature (p+r, q −r)where 0≤r≤q. Such a signature can be obtained by taking, for instance, ν1,··· , νp>1, µ1,··· , µr<0and 0< µr+1,··· , µq<1. Consider then the new first integral for the Lax-pair system given by J=I2−I1. More precisely, J(z) = (ν1−1)z2 1+···+ (νp−1)z2 p+ (1 −µ1)w2 1+···+ (1 −µp)w2 p which, with our choices, is clearly a quadratic positive definite first integral. This implies that the integral curves are all contained in a compact part of Rnand the associate geodesics are, therefore, complete curves. For metrics with signature (p−r, q +r)where 0≤r≤pthe proof follows by an analogous argument or, simply, by symmetry by replacing the metric qwith −q.□ Notice that, in particular, this proves Theorem 1.1 presented in the Introduction section. 16 A. ELSHAFEI, A.C. FERREIRA & H. REIS The leaves of F∞are then contained in the level sets of I, which naturally satisfy (14) (1 −Kν1)x2 1+ (1 −Kν2)x2 2= 1 −Kν3, for K∈R. Each leaf is then contained in a certain conic and the type of the conic in the present coordinates depends on the value of K. To completely describe the leaves of F∞on ∆∞≃RP(2), two more charts should be considered. So, let (y1, y2)and (u1, u2)be affine coordinates related with (x1, x2)as follows ψ(y1, y2) = y1 y2 ,1 y2= (x1, x2) ϕ(u1, u2) = 1 u1 ,u2 u1= (x1, x2) A vector field representing F∞in the affine coordinates (y1, y2)is then given by Y∞=y2(a−by2 1)∂ ∂y1 +y1(c−by2 2)∂ ∂y2 while, in the affine coordinates (u1, u2)is given by U∞=u2(c−au2 1)∂ ∂u1 +u1(b−au2 2)∂ ∂u2 . The foliation F∞induced by the Euler-Arnold vector field on ∆∞has then a total of 7 singular points, namely p1, p2, p3and p4given, respectively, in the affine coordinates (x1, x2, x3)by pa/c, pb/c, 0,−pa/c, pb/c, 0,−pa/c, −pb/c, 0 pa/c, −pb/c, 0. and qx, qyand qustanding, respectively, for the origin of the affine coordinates (x1, x2),(y1, y2)and (u1, u2). The dynamics of F∞on each one of the affine coordinates is summarized in the picture below. Dynamics on (x1, x2)Dynamics on (y1, y2)Dynamics on (u1, u2) Recall just that the “cone above” every leaf of F∞is invariant by F. The dynamics of the foliation over these invariant cones play a role in the proof of the proposition below, where the maximal domain of definition of every single geodesic is given. Proposition 6.4. Let Lbe a leaf of Fand L∞its projection on ∆∞. The geodesic associated with Lis complete if its projection coincides with one of the singular points qx, qyor quand is incomplete in all the other cases. An incomplete geodesic is, in fact, R+or R−complete if and only if L∞joins a singular point pito a singular point qjor if it reduces to one of the points pi. Proof. Recall that the leaves of F∞are contained in the level sets of the function Ion (13) and, consequently, their points satisfy Equation (14) for some K, which means that they are contained in a certain conic in the affine coordinates (x1, x2, x3). Let us first discuss the case where the projection of a leaf Lof Fcoincides with one of the singular points of F∞. Note that the argument to prove that the straight line “above” p1is an incomplete GEODESIC COMPLETENESS ON LIE GROUPS 17 geodesic can also be applied to the straight lines “above” the singular points p2, p3and p4. The restriction of the Euler-Arnold vector field to the mentioned leaves is a (non-zero) constant multiple of the vector field x2 3∂/∂x3so that the geodesics are, in fact, R+or R−complete. In turn, the straight line “above” the singular point qx, given by l≃ {x1=x2= 0}, is contained in the singular set of F. The restriction of Xto lvanishes identically and, therefore, the geodesic passing through every single point of lreduces to the point itself and hence it is complete. The same holds for the straight line “above” the singular points qyand qu. Next, note that every single leaf of F∞goes from one singular point of F∞to another singular point of F∞. Furthermore, since the vector field is homogeneous, Fhas no singular points away from the above mentioned straight lines. Thus, the only chance for a leaf Lof Fon R3(i.e. a geodesic) to approach infinity, is when its projection on ∆∞approaches one of the 7singular points. Therefore, in order to study the completeness of the remaining geodesics, it is sufficient to study the domain of definition of the solutions nearby each one of the 7singular points. In fact, it will be sufficient to consider the behavior of the solutions nearby the singular points p1and qxsince the pull-back of Xby rotation of angle πon the variables (x1, x2)or by reflection on the coordinate axes of affine coordinates (x1, x2)coincides with Xor with −X, respectively. Furthermore, the expressions of X∞, Y∞and U∞nearby the origin ensure as well that the behavior nearby qxis similar to the behavior nearby qyand qu. So, let us fix once and for all the affine coordinates (x1, x2, x3)nearby ∆∞and let us study the behavior and the domain of definition of the geodesics accumulating at p1 and/or at qx. With respect to px, there are just four leaves of F∞accumulating at the point in question, namely the straight lines joining qxto each one of the singular points p1, p2, p3and p4. In turn, from the symmetries previously described, it suffices to consider the leaf L∞joining qxto p1. This leaf is parameterized by H0(x1) = (x1,pb/a x1,0),x1∈]0,pa/c[, so that a leaf Lon the cone above L∞ is parameterized by H(x1)=(x1,pb/a x1, x3(x1)) where x3satisfies dx3 dx1 =−cx1 a−cx2 1 x3so that x3(x1) = x0 3sa−cx2 1 a−c(x0 1)2. Thus, for Ldistinct from L∞, the absolute value of x3=x3(x1)is bounded from below by a strictly positive constant nearby x1= 0. Thus, the associated geodesic remains away from ∆∞(it goes to a singular point of Fon the straight line “above” qx) so that it is, at least, R+or R−complete, depending on weather the geodesic is approaching or moving away from the singular point as t increases. To prove that the geodesic is, in fact, incomplete we go as follows. First, note that x3=x3(x1)goes to zero when x1goes to the singular point pa/c, which means that Lapproaches ∆∞. Next, consider the time-form associated with Xalong L, which is given by dT =x3(x1) x2(x1)(a−cx2 1)dx1=C x1p|a−cx2 1|dx1, for some constant C∈R∗. The time it takes to go from a point (x0 1, x0 2, x0 3)to p1along Lis given by R√a/c x0 1 dT and so, to decide whether the geodesic reaches ∆∞in finite time or not, we just need to decide on the convergence or divergence of the mentioned integral. Since a−cx2 1may be written as a(1 −pc/a x1)(1 + pc/a x1)and since R√a/c x0 1 1 q1−√c/a x1 dx1converges, there follows that R√a/c x0 1 dT converges as well and the corresponding geodesic is incomplete. From now on let us focus on the foliation nearby p1. We claim first that the geodesics associated with the leaves in the invariant plane {x2=pb/a x1}are R+or R−complete. In fact, this is the case for those where x1∈]0,pa/c[. As for the remaining leaves, the claim follows from similar calculations along with the previously mentioned symmetry arguments. Consider then the invariant plane {x2=pb/c}and assume first that L∞is such that −pa/c < x1<pa/c. If Lis a leaf of F 18 A. ELSHAFEI, A.C. FERREIRA & H. REIS away from ∆∞and projecting on L∞, then the variable x3satisfies dx3 dx1 =−cx1 a−cx2 1 x3so that x3(x1) = x0 3sa−cx2 1 a−c(x0 1)2. We have that x3goes to zero as x1goes to ±pa/c, which means that Lapproaches ∆∞. Let us then check that the associated geodesic reaches infinity in finite time, so that it is incomplete. The time-form associated with Xalong Lis given by dT =x3(x1) x2(x1)(a−cx2 1)dx1=C p|a−cx2 1|dx1, for some constant C∈R∗. Again, the time it takes to go from the point (x0 1, x0 2, x0 3)to p1along Lis given by R√a/c x0 1 dT. Again, this integral converges if and only if so does R√a/c x0 1 1 q1−√c/a x1 dx1and, since the later is clearly convergent, the associated geodesic is incomplete. Taking into account the previously described symmetries, the geodesic is neither R+nor R−complete. In other words, the maximal interval of definition is a bounded open interval. The study of the remaining leaves on the invariant plane {x2=pb/c}is similar, with the exception of the fact that their projection onto ∆∞escapes from every compact subset of the affine chart (x1, x2). If Lis a leaf on this plane with x1>pa/c, then the argument above allows us to say that Lreaches ∆∞in finite time when its projection approaches the singular point p1. In turn, to discuss R+or R−completeness of the leaf, either we consider the affine charts (w1, w2, w3)related with (z1, z2, z3)through the map (15) Λ(w1, w2, w3) = 1 w1 ,w2 w1 ,w3 w1= (z1, z2, z3), and where the divisor at infinity is represented by {w1= 0}, or we study directly the time-form in the present coordinates. Following the second approach, we may note that Z+∞ x0 1 dT =Z+∞ x0 1 C p|a−cx2 1|dx1=Z1/x0 1 0 C sp|as2−c|ds clearly diverges since p|as2−c|is bounded in the considered interval. The leaf Lis then R+or R−complete. The study of the leaves on the invariant plane {x1=pa/c}is analogous. Assume now that Lis a leaf whose projection L∞is contained in an ellipse and accumulates at p1. The leaf L∞can thus be parameterized by H(θ) = (α1cos θ, α2sin θ), where αi=q1−Kν3 1−Kνi, i=1,2, for some Ksuch that all 1−Kνi,i= 1,2,3, have the same sign. Furthermore, we can assume that θbelongs to ]θ1, θ2[, where θiis such that H(θi) = pi. The cone Sabove L∞then admits a natural parametrization by (θ, x3)and the pull-back of Xthrough this parametrization is given by 1 x3−aα2 α1 +cα1α2cos2θ∂ ∂θ −x3cα1α2cos θsin θ∂ ∂x3. Thus, it can easily be checked that x3(θ) = x0 3s−aα2+cα2 1α2cos2θ −aα2+cα2 1α2cos2θ0 . The height function |x3|=|x3(θ)|goes to zero as θgoes to θ1(and to θ2as well), which means that Lapproaches ∆∞. The time-form over Lis given by dT =C p|−aα2+cα2 1α2cos2θ|dθ , for some C∈R∗(naturally assuming Ldistinct from L∞). Now, we have the following. GEODESIC COMPLETENESS ON LIE GROUPS 19 Claim: There exists ε, with 0< ε < θ2−θ1, and M > 0such that 0≤M|θ−θ1| ≤ −aα2+cα2 1α2cos2θ for every θ∈]θ1, θ1+ε[. Proof of the Claim. Consider the function u(θ) = −aα2+cα2 1α2cos2θ. Since uvanishes at θ=θ1, the claim immediately follows if u′(θ1)= 0. In fact, if this is the case, it suffices to take M= |f′(θ1)|/2. Since u′(θ) = −2cα2 1α2sin θcos θand recalling that α1cos θ1=pa/c and α2sin θ1= pb/c, we get that u′(θ1)=2α1√ab. The value of u′(θ1)is clearly non-zero and the result follows. □ As an immediate consequence of the previous claim, we have Zθi+ε θi C p|−aα2+cα2 1α2cos2s|ds ≤Zθi+ε θi C pM|s−θ1|ds < ∞. In other words, the corresponding geodesic reaches infinity in finite time and hence the geodesic is incomplete. In fact, there follows from the previously mentioned symmetries that the maximal domain of definition is an open interval. The case where Lis a leaf whose projection L∞is part of a hyperbola accumulating at p1is left to the reader - the calculations for the case of an ellipse apply equally well to this case. □ Let us finish this section by making some comments on the dynamics of Fwhen ab < 0. Recall that, when ab < 0, the leaves of Fare contained in a compact part of R3. This is in contrast with the case where ab > 0, were geodesics escaping from every compact subset of R3naturally appear associated with the so-called idempotents. In turn, an idempotent induces a singular point for the foliation induced in ∆∞. In the case where ab < 0we have no idempotents. In fact, if c= 0, there are only 3 singular points for F∞, namely the origin of each one of the three charts considered above. Each straight line “above” them, although invariant for F, is constituted by singular points of F. If c= 0, the origin of the affine coordinates is the only singular point for Fand the straight line “above” it is also constituted by singular points. Note that in the case where c= 0, the leaves of F∞in the affine coordinates (x1, x2)are nothing but circles centered at the origin and the line at infinity of the mentioned chart is a leaf as well. In particular, the foliation has no singular points on the other charts. The leaves on the invariant cones are all closed from Proposition 6.3. As for the case where c= 0 1. the foliation F∞has a center at the origin of the affine coordinates (x1, x2); 2. the foliation F∞has a center at the origin of the affine coordinates (y1, y2)and a saddle at the origin of the affine coordinates (u1, u2)in the case where ac < 0; 3. the foliation F∞has a saddle at the origin of the affine coordinates (y1, y2)and a center at the origin of the affine coordinates (u1, u2)in the case where ac > 0; The figure below exhibits the leaves of F∞in the case 0< ν1< ν2< ν3where, in particular, we have ac < 0. Dynamics on (x1, x2)Dynamics on (y1, y2)Dynamics on (u1, u2) Again, the leaves over each invariant cone are closed. 20 A. ELSHAFEI, A.C. FERREIRA & H. REIS 7. Characterization of geodesics in case 3 Let us now consider the case where the isomorphism Φadmits a (real) eigenvalue λsuch that ma(λ)−mg(λ)=1. It has been proved in Lemma 5.3 that there exists a B-pseudo-orthonormal basis v= (vk), satisfying B(v1, v1) = B(v2, v3)=1and B(v1, vk) = B(vk, vk) = 0 for k= 2,3, with respect to which the isomorphism Φcan be written as in Section 5, where ζ= 0 coincides with q(v3, v3). In this case Φ−1is given by Φ−1= η0 0 0ν−ζν2 0 0 ν  (16) where η= 1/µ and ν= 1/λ. As previously done, fix an element z∈sl(2,R)and let (z1, z2, z3) stand for its coordinates with respect to the basis v= (vk). From Lemma 5.2 there follows that the Euler-Arnold vector field is written, in the affine coordinates (z1, z2, z3)∈R3, as E=ζν2z2 3 ∂ ∂z1 +z1(ν−η)z2−ζν2z3∂ ∂z2 + (η−ν)z1z3 ∂ ∂z3 .(17) 7.1. Characterization of geodesically complete metrics. The characterization of the geodesically complete left-invariant pseudo-Riemannian metrics is given, in this case, not only in terms of the eigenvalues of Φ(or, equivalently, Φ−1) but also in terms of ζ=q(v3, v3). To be more precise, the following can be said. Theorem 7.1. The left-invariant pseudo-Riemannian metric qis geodesically complete if and only if (η−ν)ζ≤0. To prove Theorem 7.1 we may consider separately the cases where (η−ν)ζ > 0and (η−ν)ζ≤0. As in the previous Section, we will prove the existence of the so-called idempotents in the case where (η−ν)ζ > 0. As to the case where (η−ν)ζ≤0the proof has to be different from the corresponding one where Φis diagonalizable. In fact, we will see that in this case the leaves although complete are not contained in a compact subset of R3. The proof of the theorem is then divided into the two following propositions. Proposition 7.2. If (η−ν)ζ > 0then there exists at least one geodesic that is incomplete. Proof. Consider the rational extension of the Euler-Arnold vector field to RP(3). The plane at infinity ∆∞is invariant by the foliation Finduced by this extension since the Euler-Arnold vector field is a homogeneous polynomial vector field distinct from a multiple of the radial vector field. Next, consider the affine coordinates (x1, x2, x3)nearby ∆∞related to (z1, z2, z3)through the map Ψdefined by (10). The Euler-Arnold vector field is given, in the coordinates (x1, x2, x3), by (18) X=1 x3(b−ax2 1)∂ ∂x1−x1(b+ 2ax2)∂ ∂x2−ax1x3 ∂ ∂x3, where a=η−νand b=ζν2. The assumption on the statement of the proposition translates, in these new parameters, by ab > 0and it can easily be checked that the intersection of the singular set of Fwith ∆∞, in the considered affine chart, is constituted by 2points, namely p1= rb a,−b 2a,0!and p2= −rb a,−b 2a,0!. The straight lines “above” each one of these singular points are regular leaves for F. In fact, the restriction of Xto each of these straight lines is a constant vector field. For example, the restriction of Xto the straight line (or, more precisely, leaf L) above p1is given by XL=−apb/a ∂ ∂x3. In particular, XLis regular at the origin, which implies that Lreaches the plane at infinity in finite time. The corresponding geodesic is therefore incomplete and the result follows. □ Let us now consider the case where ab ≤0. In this case the following can be proved. GEODESIC COMPLETENESS ON LIE GROUPS 21 Proposition 7.3. If ab ≤0then all geodesics are complete. Proof. The proposition can be easily proved in the case where ab = 0. In fact, assuming ab = 0 (or, equivalently, a= 0 since b= 0 by assumption), Φhas a unique eigenvalue, whose difference between its algebraic and geometric multiplicity is equal to 1. The third component of Xin (17) vanishes identically and, hence, the Euler-Arnold vector field reduces to bk2∂ ∂z1−bkz1∂ ∂z2, which is clearly complete since the solution of the associated differential system is polynomial with respect to t. Assume now that ab < 0. The first integrals I1(z) = B(z, z)and I2(z) = B(z, Φ−1z)for Fare given in coordinates (z1, z2, z3)by I1(z1, z2, z3) = z2 1+2z2z3and I2(z1, z2, z3) = ηz2 1−ζν2z2 3+2νz2z3, which are clearly linearly independent since ζ= 0. Consider then the linear combination of the present first integrals given by I=νI1−I2 I(z1, z2, z3) = (ν−η)z2 1+ζν2z2 3=−az2 1+bz2 3, and which is also a first integral for F. Note that Iis a quadratic positive definite first integral for the system in the variables z1and z3formed by the first and third equations associated with the Euler-Arnold system. This means that the projection of every leaf of Fthrough the map π(z1, z2, z3) = (z1, z3)is contained in a compact part of R2(although the leaf itself may escape to infinity). This means that every solution of the Euler-Arnold system is such that the expression of (z1, z3)=(z1(t), z3(t)) is defined for every t. The completeness of the solution (z1, z2, z3) = (z1(t), z2(t), z3(t)) of the Euler-Arnold system thus depends on the second equation. This equation, in turn, can be seen as a first order non-homogeneous linear differential equation in the variable z2. In fact, it corresponds to the differential equation ˙z2=f(t)z2+g(t)where fand gare the differential functions on Rgiven, respectively, by f(t)=(ν−η)z1(t)and g(t) = −ζν2z1(t)z3(t). The general solution of such a differential equation takes on the form z2(t)=(G(t) + C)eF(t) where Fis a primitive function of f,Gis a primitive function of g(t)e−F(t)and Ca real constant. Since fand gare differentiable functions defined on Rso are their primitive functions and, consequently, so is z2=z2(t). We thus conclude that (z1, z2, z3) = (z1(t), z2(t), z3(t)) is complete. □ In what follows, we will use the method above to give an alternative proof for the characterization of the geodesically complete metrics. Furthermore, in the case of incomplete metrics, we will present the maximal domain of definition of every single geodesic. 7.2. Geodesics for ab < 0.Recall that Fis given in the affine coordinates (x1, x2, x3)by the vector field Xin Equation (18). The foliation F∞induced by Fin ∆∞is, in particular, represented by the vector field (19) X∞= (b−ax2 1)∂ ∂x1−x1(b+ 2ax2)∂ ∂x2 and it becomes clear that F∞does not admit singular points in the present coordinates. However, F∞has two singular points namely, the origin of the affine coordinates (y1, y2)and the origin of the affine coordinates (u1, u2)(with (y1, y2)and (u1, u2)as in Section 6). By calculating the vector fields Y∞and U∞, representatives of F∞in the affine coordinates (y1, y2)and (u1, u2), respectively, it becomes clear that the origin is a degenerate singular point for Y∞while the origin is a saddle for U∞. Furthermore, by taking the quotient of the two independent first integrals I1, I2on the affine coordinates (x1, x2, x3), it is clear that the leaves of F∞are all contained in conics. The figures below represent F∞in the different affine coordinates. 22 A. ELSHAFEI, A.C. FERREIRA & H. REIS Dynamics on (x1, x2)Dynamics on (y1, y2)Dynamics on (u1, u2) By using the quotient of the two first integrals I1and I2, it can easily be checked that the leaves of Fover ∆∞are given in coordinates (x1, x2)by x2=αx2 1+β, with α=Kη−1 2(1−Kν)and β=−Kζν2 2(1−Kν), which means that the leaves can be globally parameterized through the variable x1. Fix then a leaf L∞of Fover ∆∞and note that the cone above L∞can naturally be parameterized through (x1, x3). The pull-back of the Euler-Arnold vector field to the mentioned cone is given by 1 x1(b−ax2 a)∂ ∂x1−ax1x3 ∂ ∂x3. Fixed a leaf Lin the mentioned cone (and naturally away from ∆∞), we have that along L (20) dx3 dx1 =−x1 b−ax2 1 x3so that x3=x0 3sb−ax2 1 b−ax0 12. Now, since the time-form of the vector field along Lis given by dT =x3(x1) b−ax2 1 dx1=C p|b−ax2 1|dx1 for some non-zero real constant C, we can easily conclude that the integral of the time-form along the entire leaf diverges. In fact, since Z+∞ 1 C p|b−ax2 1|dx1=Z+∞ 1 C x1qb/x2 1−adx1 and qb/x2 1−ais bounded in the interval of integration, we get that the latter integral diverges. Furthermore, since the integrand is an even function, the integral of the time-form along the entire leaf diverges as well. We thus conclude that the geodesic associated with Lis complete. Let us finally consider the leaves on the cone above the line at infinity of the affine coordinates (x1, x2), which is invariant by F∞. To study the mentioned leaves, we should then consider the affine coordinates (w1, w2, w3)related with (z1, z2, z3)through the map Λon R3\{w1= 0}defined by (15), where divisor at infinity is given by {w1= 0}and the invariant plane by {w3= 0}. The Euler-Arnold vector field given in these coordinates by W=1 w1−bw1w2 3 ∂ ∂w1−(aw2+bw3+bw2w2 3)∂ ∂w2 +w3(a−bw2 3)∂ ∂w3 so that its restriction to the invariant plane {w3= 0}is simply given by −aw2/w1∂/∂w2. The variable w1along every single leaf Lin the mentioned plane is such that dw1 dw2≡0and, consequently, w1is constant for all t. This implies that Lremains way from ∆∞nearby the origin of the present coordinates. The leaf is therefore, at least, R+or R−complete. To decide on the completeness of the leaf we should take the integral of the associated time-form along the part of the leaf escaping from the domain of these charts. GEODESIC COMPLETENESS ON LIE GROUPS 23 Recall that Lis parameterized by H(w2)=(c, w2,0) for some (non-zero) real constant cand w2>0or w2<0. Assume, for simplicity, that w2>0along L. Since dw2/dT =−aw2/w1, the time-form takes on the form dT =−c aw2 dw2 It becomes clear that R+∞ 1dT diverges and, consequently, the geodesic associated with Lis complete. 7.3. Geodesics for ab > 0.Proposition 7.2 ensures the existence of incomplete geodesics in the case where ab > 0. To be more precise, it was proved that the straight line transverse to ∆∞and passing through p1is associated with incomplete geodesics. Let us now investigate the domain of definition of the remaining geodesics. Consider the restriction of Fto the divisor at infinity ∆∞along with the affine coordinates (x1, x2),(y1, y2)and (u1, u2)previously introduced. By considering the representative X∞of the mentioned foliation (cf. Equation (19)) in the affine coordinates (x1, x2), it can easily be checked that F∞possesses four singular points: two singular points in the affine coordinates (x1, x2)(that, since distinct from the origin of these coordinates, they are also presented in the other two affine coordinates) along with the origins of (y1, y2)and of (u1, u2). As previously mentioned, the quotient of the first integrals I1and I2defines a first integral for F∞and, since I1, I2are homogeneous polynomials of degree 2, the leaves of F∞are conics in the different affine coordinates for ∆∞. The figure below represents F∞in the different affine coordinates. Dynamics on (x1, x2)Dynamics on (y1, y2)Dynamics on (u1, u2) The characterization of the domain of definition of the geodesics for ab > 0are described in the proposition below. Proposition 7.4. Consider a leaf Lof Fand let L∞stand for its projection onto ∆∞. The geodesic associated with Lis complete if L∞is contained in the line at infinity of the affine coordinates (x1, x2) and incomplete in the other cases. An incomplete geodesic is, in fact, R+or R−complete if L∞is not contained in the bundle defined by −pb/a < x1<pb/a or, in other words, if L∞is not a leaf joining the singular points p1and p2. Proof. Recall that if L∞is a leaf of F∞contained in the chart associated with (x1, x2), then its points satisfy the equation (1 −Kη)x2 1+ 2(1 −Kν)x2=−Kζν2for some K∈R. In particular, the leaves can be globally parameterized by the variable x1with the exception of two that correspond to the vertical straight lines given by x1=±pb/a. Assume first that L∞is one of the parabolas described above. We proceed as in the case ab < 0. More precisely, we start by noticing that a leaf Lin the cone above L∞can be parameterized as H(x1) = (x1, x2(x1), x3(x1)), where x2(x1) = x2=Kη −1 2(1 −Kν)x2 1−Kζν2 2(1 −Kν)and x3(x1) = x3=x0 3sb−ax2 1 b−ax0 12, since x3satisfies the differential equation (20). Suppose that Lis such that the domain of definition of the parametrization above is the bounded interval ]−pb/a, pb/a[. In this case, we have that L 24 A. ELSHAFEI, A.C. FERREIRA & H. REIS approaches the plane at infinity either when x1approaches −pb/a or pb/a. The time needed to go through Lis given by Z√b/a −√b/a dT =Z√b/a −√b/a C p|b−ax2 1|dx1. The denominator of the integrand is given by rqb a−x1rqb a+x1, up to a multiplicative constant, so that we conclude that the integral in question converges. The associated geodesic is then incomplete, being defined in a bounded interval. Assume now that the domain of the parametrization for Lis ]pb/a, +∞[. It is easy to check that the geodesic associated with Lis incomplete. In fact, from the expression of x3=x3(x1) we know that Lapproaches ∆∞. Indeed, as previously seen, x3goes to zero as x1goes to pb/a. Furthermore, the argument presented above allows us to claim that R√b/a+ε √b/a dT converges for every ε > 0. We then reach ∆∞in finite time. The geodesic is, however, R+or R−complete. In fact, it can easily be checked that R+∞ √b/a+εdT diverges. Note that the argument presented in the case ab < 0applies equally well in this case and the other case can be treated similarly. Consider now the restriction of Xto the invariant plane {x1=pb/a}(the restriction to {x1= −pb/a}is analogous), which is given in the present coordinates by 1 x3"−rb a(b+ 2ax2)∂ ∂x2−arb ax3#. It can easily be checked that x3is given by x3(x2) = x0 3sb+ 2ax2 b+ 2ax0 2 . The expression of x3allows us to say that every leaf Lon the mentioned plane approaches ∆∞ when its projection L∞approaches the singular point p1. The leaf L∞then leaves the domain of the present chart and accumulates at the origin of the affine coordinates (y1, y2). Since the affine coordinates (x1, x2, x3)parameterize the entire leaf L, we will use them to estimate the integral of the time-form along the leaf. Note that the time-form is given by dT =−x3(x2) qb a(b+ 2ax2) dx2=C p|b+ 2ax2|dx2 for some C∈R∗. It is clear that the integrals R−b 2a+ε −b 2a dT and R+∞ −b 2a+εdT,ε > 0, have different nature. In fact, the first one converges while the second one diverges so that the associated geodesics are R+or R−complete. It remains to consider the leaves contained in the invariant cone above the line at infinity of the affine coordinates (x1, x2). The line in question is given in the coordinates (u1, u2)by u1= 0. Consider then the affine coordinates (w1, w2, w3)related with (z1, z2, z3)through the map Λdefined by (15). In the present coordinates the divisor at infinity is given by {w1= 0}and (w3, w2)coincides with (u1, u2). The proof that the leaves in the invariant plane {w3= 0}are complete goes word by word as in the case ab < 0in Subsection 7.2. In fact, the calculations involved do not depend on the parameters aand b.□ 8. Characterization of geodesics in cases 2 and 4 We will prove that every left-invariant pseudo-Riemannian metric in cases 2 and 4 is geodesically incomplete by exhibiting an incomplete geodesic for each one of these metrics. As in the previous cases, we will also characterize the maximal domain of definition of every single geodesic. GEODESIC COMPLETENESS ON LIE GROUPS 25 8.1. Case 2. In the case where the isomorphism Φhas two non-real eigenvalues, there exists a B-orthonormal basis v= (vk)with respect to which the isomorphism Φtakes on the form presented in Section 5, with β= 0. The isomorphism Φ−1is then given by Φ−1= η0 0 0γ ζ 0−ζ γ  where η= 1/µ,γ=α/(α2+β2)and ζ=−β/(α2+β2). Since we are assuming β= 0, we have ζ= 0 as well and, in this particular case, the following can be said. Theorem 8.1. There exists at least one geodesic that is incomplete. Proof. Let (z1, z2, z3)∈R3stand for the coordinates of z∈sl(2,R)in the above mentioned Borthonormal basis v. In the present coordinates, the Euler-Arnold vector field is given by (21) E=b(z2 2+z2 3)∂ ∂z1 +z1(az3−bz2)∂ ∂z2 +z1(az2+bz3)∂ ∂z3 , where a=γ−ηand b=ζ(and, in particular, b= 0). Consider now the usual affine coordinates (x1, x2, x3)where ∆∞≃ {x3= 0}. The vector field Eis written in these coordinates as X=1 x3b(x2 2−x2 1+ 1) −ax2 1x2∂ ∂x1 +x1a(1 −x2 2)−2bx2∂ ∂x2−x1x3(ax2+b)∂ ∂x3 and the induced foliation on this chart has two singular points p1, p2over ∆∞. The position of the singularities depend, however, on the parameters aand b. More precisely, in the case where a= 0, the two singular points on ∆∞are p1= (1,0,0) and p2= (−1,0,0). In turn, in the case where a= 0, we have that p1= (ρ1, ρ2,0) and p2= (−ρ1, ρ2,0), where ρ1=q2b(−b+pb2+a2)/|a|and ρ2= (−b+pb2+a2)/a , if b > 0, and ρ1=q2b(−b−pb2+a2)/|a|and ρ2= (−b−pb2+a2)/a if b < 0. It can easily be checked that the restriction of Xto the line above each one of the mentioned singular points is a (non-zero) constant vector field meaning that the associated geodesic reaches ∆∞in finite time. These geodesics are then incomplete. □ The study of this case is arduous since, as becomes clear from the proof of Theorem 8.1, not only the cases a= 0 and a= 0 should be considered separately; also in the case where a= 0, we should look separately at the cases b > 0and b < 0. We will describe the foliation for the generic case where a= 0 and b > 0and the figure below exhibits the leaves of F∞in the case where 0< η < γ < ζ. Dynamics on (x1, x2)Dynamics on (y1, y2)Dynamics on (u1, u2) The foliation F∞has a total of three singular points, namely the two singular points p1, p2 mentioned above along with the origin of the affine coordinates (u1, u2), the latter being a saddle for the mentioned foliation. It is clear from the picture that every leaf of F∞accumulates at both 32 A. ELSHAFEI, A.C. FERREIRA & H. REIS Proof of the Claim. Let σ∗E|L=F(t)∂/∂t. Owing to Riemann’s Theorem, it suffices to prove that the limit of Fat 0exists and is non-zero. Consider then local coordinates (x, y, z)nearby p1where p1≃(0,0,0) and ∆∞≃ {z= 0}. It can be easily checked that in these coordinates, the vector field Etakes on the form 1 z(2x+ h.o.t.)∂ ∂x + (2y+ h.o.t.)∂ ∂y + (z+ h.o.t.)∂ ∂z , up to a (non-zero) multiplicative constant. Thus the pull-back of the restriction of Eto Lby σis given by (1 + h.o.t) ∂ ∂t up to the same multiplicative constant. This means that σ∗E|Lis regular at 0∈Cas we intended to prove. □ The normalization b Lof the compactification of Lin CP(3) is a (compact) Riemann surface equipped with a globally defined holomorphic vector field. This vector field if therefore complete on b L. Since the restriction of Eto Lis identified with the restriction of the mentioned vector field to the complement of finitely many points in b L, it must be semicomplete as the restriction of a complete vector field to an open set. This holds for an arbitrary leaf of F, we conclude that Eis semicomplete on C3. Consider next the case where Φhas an eigenvalue λsuch that ma(λ)−mg(λ)=1and consider aB-pseudo-orthonormal basis where Φ−1takes on the form (16), so that the Euler-Arnold vector field is written as (17) E=bz2 3 ∂ ∂z1−z1(az2+bz3)∂ ∂z2 +az1z3 ∂ ∂z3 , where a=η−νand b=ζν2. As previously seen, if Φhas an eigenvalue whose eigenspace has dimension at least 2, i.e. if a= 0, then the Euler-Arnold vector field is complete on C3. Thus it is semicomplete on C3as well. Assume then that a= 0. Given the preceding construction, it suffices to show that whenever a leaf Lof Finduces a separatrix for a singularity of Flying in ∆∞, the restriction of Xto the resulting separatrix is holomorphic. In the present case, Fpossesses four singular points in ∆∞, namely (i) p1=pb/a, −b/(2a),0and p2=−pb/a, −b/(2a),0in the affine coordinates (x1, x2, x3). The eigenvalues of Fat these points are 2,2,1, where 1is the eigenvalue associated to the direction transverse to ∆∞. These points can be treated exactly as the points with the same eigenvalues in the previous case. (ii) The intersection of ∆∞with the axis z1, denoted by p3. The eigenvalues of Fat these points are 1,−1,0, where 0is the eigenvalue associated to the direction transverse to ∆∞. In particular, the foliation induced on ∆∞has a saddle behavior. This implies that the leaves in the invariant planes defined by the two separatrices though p3are the only leaves that can accumulate at p3. By expressing the Euler-Arnold vector field in the affine coordinates, it can easily be checked that a leaf Lin these invariant planes never accumulates at p3unless it is totally contained in ∆∞. Summarizing, no leaf of Finduces a separatrix at p3so that this singular point plays no further role in the present discussion. (iii) The intersection of ∆∞with the axis z2, denoted by p4. The foliation has a singular point of order 2at p4. In particular all eigenvalues are zero. This case is discussed in detail below. Consider the affine coordinates (v1, v2, v3)where (v1/v2,1/v2, v3/v2)=(z1, z2, z3). The first integrals I1and I=νI1−I2(cf. Section 7) in these coordinates are given by I1=v2 1+ 2v3 v2 2 and I=−av2 1+bv2 3 v2 2 . From I1, we have that v3is a function of v1and v2. In fact, v3= (kv2 2−v2 1)/2so that the corresponding invariant surfaces for Fpass through the origin for every k∈Cand they are tangent GEODESIC COMPLETENESS ON LIE GROUPS 33 to the plane {v3= 0}at the point in question. In turn, by substituting v3=v3(v1, v2)we can check that, indeed, the leaves pass through the origin and that they are transverse to ∆∞(locally given by {v2= 0}). The leaf Ladmits then a Puiseux parametrization of the form σ(t)=(t, αt + h.o.t, t2+ h.o.t.), for some α∈C∗. Noticing that the Euler-Arnold vector field in the affine coordinates (v1, v2, v3)is given by 1 v2av2 1+bv3(v2 1+v3)∂ ∂v1 +v1v2(a+bv3)∂ ∂v2 +v1v3(2a+bv3)∂ ∂v3, the pull-back of the restriction of Eto Lby σcan be holomorphically extended to 0∈Cas a vector field of the form F(t)∂/∂t, with F(0) = 0 and F′(0) = 0. We then conclude that Lmay be compactified as an algebraic Riemann surface equipped with a global holomorphic vector field, so that it is complete. Finally, consider case (c) and the B-pseudo-orthonormal basis with respect to which the isomorphism Φtakes on the form presented in Section 5. If (z1, z2, z3)∈C3stands for the coordinates of z∈sl(2,R)in the mentioned B-pseudo-orthonormal basis, the Euler-Arnold vector field is given in these coordinates by (23) E=ζν2z3(−ζνz3+z1)∂ ∂z1 + (z2z3−z2 1+ζνz1z3)∂ ∂z2−z2 3 ∂ ∂z3. We keep the preceding notations so that Fstands for the corresponding singular holomorphic foliation in CP(3). The singular set of Fconsists of (i) an isolated singular point p∈∆∞given in the affine coordinates (x1, x2, x3)by p= ζν 2,−ζ2ν2 8,0. The eigenvalues of Fat pare 2,2,1, where 1is the eigenvalue associated to the direction transverse to ∆∞. This point can be treated exactly as the points with the same eigenvalues in the previous cases. (ii) the projective line arising from the z2-axis. The intersection point of this projective line with ∆∞will be denoted by qand coincides with the origin of the affine coordinates (v1, v2, v3). We detail below how to treat the leaves Lof Fyielding a (local) separatrix for Fat q. In the coordinates (v1, v2, v3)the foliation Fis determined by the (two independent) first integrals I1=v2 1+ 2v3 v2 2 and I2=νI1+ζ2ν3v2 3−2ζν2v1v3 v2 2 . In particular, if Lis a leaf of Fwhose projection onto ∆∞approaches q, then the actual leaf Lhas to approach q. Following the same argument used in case (b), a separatrix induced by Lis smooth at qand admits an irreducible Puiseux parametrization of the form σ(t) = (t, αt + h.o.t, t2+ h.o.t), for some α∈C∗. Finally, by noticing that the Euler-Arnold vector field in the present coordinates takes on the form V=ba2 v2(v3 1−abv2 1v3−abv2 3)∂ ∂v1 +v2(v2 1−v3−abv1v3)∂ ∂v2 +v3(v2 1−2v3−abv1v3)∂ ∂v3 there follows that σ∗V|Ladmits a holomorphic extension to 0∈Cof the form F(t)∂/∂t, for some Fsuch that F(0) = F′(0) = 0 and F′′(0) = 0. Again, this shows that the restriction of the EulerArnold vector field to the normalization of the compactification of Lis holomorphic and ends the proof of the theorem. □ Acknowledgements. The authors are grateful to Julio Rebelo for providing us with a copy of his unpublished manuscript [23] and would also like to thank Ilka Agricola, Miguel Sánchez and Abdelghani Zeghib for their valuable comments on this manuscript. The first author was financed by FCT - Fundação para a Ciência e Tecnologia, I.P. (Portugal) - through the PhD scholarship PD/BD/143019/2018. 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Springer, vol. 73 (2015), 343–372. [27] N. Tholozan, Uniformisation des variétés pseudo-riemanniennes localement homogènes, PhD. Thesis, Univ. NiceSophia Antipolis (2014). Ahmed Elshafei, Centro de Matemática da Universidade do Porto, Centro de Matemática da Universidade do Minho, Portugal, [email protected] Ana Cristina Ferreira, Centro de Matemática da Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal, [email protected] Helena Reis , Centro de Matemática da Universidade do Porto, Faculdade de Economia da Universidade do Porto, Portugal, [email protected]