Rigidity of Holomorphically Projective Mappings of Kähler Spaces with Finite Complete Geodesics

Citation: Vítková, L.; Hinterleitner, I.; Mikeš, J. Rigidity of Holomorphically Projective Mappings of Kähler Spaces with Finite Complete Geodesics. Mathematics 2024,12, 1239. https:// doi.org/10.3390/math12081239 Academic Editors: Ion Mihai and Adela Mihai Received: 31 March 2024 Revised: 14 April 2024 Accepted: 15 April 2024 Published: 19 April 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Rigidity of Holomorphically Projective Mappings of Kähler Spaces with Finite Complete Geodesics Lenka Vítková 1,∗, Irena Hinterleitner 2and Josef Mikeš 1 1Department of Algebra and Geometry, Faculty of Science, Palacký University in Olomouc, 779 00 Olomouc, Czech Republic; [email protected] 2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic; hinterleitner[email protected].cz *Correspondence: [email protected] Abstract: In this work, we consider holomorphically projective mappings of (pseudo-) Kähler spaces. We determine the conditions for finite complete geodesics that must be satisfied for the mappings to be trivial; i.e., these spaces are rigid. Keywords: geodesic; holomorphically projective mappings; Kähler space; rigidity; Riemann tensor; symmetric space MSC: 53C24; 53B35; 53C15; 53C22 1. Introduction This work develops some new ideas in the theory of holomorphically projective mappings of Kähler spaces. These questions are connected with the compact and complete geodesics, Kähler spaces, and their holomorphically projective mappings and transformations. In 1954, Westlake [ 1 ] and Yano [ 2 ] studied the geodesic mappings of Kähler spaces. They proved that if the structure of the Kähler space is preserved, then the mapping is trivial. This result was generalized by Muto [ 3 ], for the case where these structures commute. Mikeš proved that geodesic mappings of Kähler spaces can exist [ 4 – 7 ], eventually onto Kähler spaces (see [ 8 – 14 ]). These Kähler spaces are equidistant (Sinyukov [ 15 , 16 ]); i.e., they admit convergent vector fields (Shirokov [ 17 – 19 ]), which are special concircular vector fields (Yano [20]). Analytically planar curves and holomorphically projective mappings of Kähler spaces introduced by Otsuki and Tashiro [ 21 ] are a natural generalization of geodesics and geodesic mappings. In these mappings, analytically planar curves are mapped onto analytically planar curves. They showed that spaces with a constant holomorphic curvature of holomorphically projective mapping have properties similar to those of spaces with a constant curvature with respect to geodesic mappings. An overview of the results up to 1963 on holomorphically projective mappings is available in Beklemishev [ 22 ], Yano, and Bochner [23,24], for example. Mikeš generalized these results for holomorphically projective mappings in different directions [ 25 , 26 ]; some of these results are included in the fifth (last) chapter of Sinyukov’s monograph [ 16 ]. These results can be found in [ 10 ] and in [ 4 , 8 , 11 – 13 ]. More results can be found in Mikeš dissertation [ 4 ], in particular, concerning Kn[B] , see Section 4. These general results were published in [10]. Other problems and ideas in the theory of holomorphically projective mappings were developed by Aminova [ 27 – 30 ] and others. The complex projective space (CP(n) , gFubini-Study) admits global non-trivial holomorphically projective mappings and transformations with maximal parameters (see [ 31 , 32 ]). Previously, locally, for spaces with constant holomorphic Mathematics 2024,12, 1239. https://doi.org/10.3390/math12081239 https://www.mdpi.com/journal/mathematics Mathematics 2024,12, 1239 2 of 13 curvature, the same was proved in [ 25 , 26 ]. These calculations were mostly performed in a complex form. The same is true in other works, e.g., [ 32 ]. In this work, Equation (3) was not attributed to Mikeš (see [4,10]). Holomorphically projective mappings of hyperbolic and parabolic Kähler spaces have been dealt with in Prvanovi´c [33], Kurbatova [14,34], and Shiha [35]. Holomorphically projective mappings views have been generalized in many ways. In 1962, A.Z. Petrov [ 36 ] studied quasi-geodesic mappings, where he showed that it is possible to simulate physical processes and electromagnetic fields. Similar results were presented in the paper of C.-L. Bejan and O. Kowalski [ 37 ]. The abovementioned mapping generalized the F -planar mappings of Mikeš and Sinyukov [ 38 ]. The almost geodesic mappings π2 are also a direct generalization of holomorphically projective mappings (see [ 16 ] and [10–13,39] ). In a 2019 paper [ 40 ] by A. Kozak and A. Borowiec, the authors studied a new physical interpretation of almost geodesic mappings that are special transformations, which genuinely preserve geodesics in space and time. The problems connected with these topics have been considered in many monographs and reviews, such as [41–48]. Many authors have dealt with rigidity problems, i.e., when the holomorphically projective mappings will be affine (trivial). We follow these works on similar problems of rigidity, which were studied for motions (Killing vector fields) and their generalization in compact or complete Riemann and Kähler spaces (see the monographs by Yano and Bochner [23,24]). Using Bochner’s methods (see Stepanov [ 49 ]), Tachibana and Ishihara [ 50 , 51 ], Hasegawa and Yamauchi [ 52 ], and Akbar-Zadeh and Couty [ 53 , 54 ] also discovered new results. Later, Sinyukov [ 55 ] and Mikeš [ 10 , 56 ] also continued this research. Due to the method of Švec [ 57 ], even more general results were found [58]. In 1961, Tachibana and Ishihara [ 59 ] proved that Ricci symmetric (non-Einstein) spaces do not admit nontrivial analytical holomorphically projective transformations. Then, in 1979, Mikeš proved that these spaces also do not admit nontrivial mappings, while global requirements are not assumed, [4,10]. See also Bácsó and Ilosvay [60]. Sakaguchi [ 61 ] used Sinyukov’s methods (see [ 15 ]) and proved that symmetric and recurrent Kähler spaces of non-constant holomorphically projective curvature do not admit non-trivial holomorphically projective mappings. Domashev and Mikeš [ 25 ] generalized Sakaguchi’s results for (pseudo-) Kähler spaces. The main results of our study are Theorems 2and 3. They clearly state that in order for the mapping to be rigid the space does not have to be complete. It suffices that there exist a finite number of geodesics and their images that are complete. In other words, the space is uniquely defined by the given geodetics, which are the supporting skeleton (reinforcement) of the space. 2. Kähler Spaces Kähler space Kn is an n dimensional (pseudo-) Riemannian space in which, along with the metric tensor g , an affine structure F is defined that satisfies the relations F2=−Id , g(X , FX) = 0, and ∇F= 0, where ∇ is the Levi–Civita connection, and X is any tangent vector on Kn . Necessarily, the spaces Kn are of an even dimension, i.e., n= 2 m , and n≥ 4. In local coordinates x≡(x1 , x2 , . . . , xn) , components gij(x) and Fh i(x) of g and F satisfy the relations Fh αFα i=−δh i;Fα (igj)α=0; Fh i,j=0. Here and in what follows, “,” denotes a covariant derivative on Kn and the round brackets denote the symmetrization of indices. The structure Fis called a complex structure. The spaces Kn were first considered by Shirokov [ 18 ]. Independently, in complex form, these spaces were studied by Kähler [ 62 ]. In the available literature, these spaces are also called Kählerian. We present the notation that is used in Mikeš’s dissertations [ 4 , 8 ] and in many articles, for example, [10–14,16,22,23]. Mathematics 2024,12, 1239 3 of 13 In the Kähler spaces Kn , we introduce the operation of the conjugation of indices as follows: A··· ··· i··· ≡A··· ··· α··· Fα i;A··· j··· ··· ≡A··· α··· ··· Fj α. According to the definition of a tensor F, this operation has the following properties: Ai=−Ai;Bi=−Bi;AαBα=AαBα;AαBα=−AαBα;(Ai),j=Ai,j;(Bi),j=Bi ,j. For the Kronecker symbol, metric, and and its inverse tensors it holds that δh i=δh i=Fh i;gij +gij =0; gi j =gij;gij +gij =0; gi j =gij. For the Riemann and Ricci tensors, Rh ijk =∂jΓh ik −∂kΓh ij +Γh αjΓα ik −Γh αkΓα ij , ∂j=∂/∂xj , and Rik =Rα iαk, the following formulas hold: Rhij k =Rhijk ≡ghαRα ijk;Ri j =Rij;Rα αjk =2Rjk. In the Kähler spaces Kn , we can consider the holomorphically projective curvature tensor Ph ijk ≡Rh ijk +1 n+2(δh kRji −δh jRki +δh kRȷi−δh ȷRki −2δh ıRȷk). When specific maps f of spaces are considered, say, Kn f →Kn , both spaces are assigned to the coordinate system x , in general, with respect to these mappings. In this coordinate system, the corresponding points x∈Kn and f(x)∈Kn have the same coordinates x≡x1,x2, . . . , xn. In this case, we denote the corresponding geometric objects in An with a bar; for instance, Rh ijk and Rij are the Riemannian and Ricci tensors. 3. General Questions Concerning Holomorphically Projective Mappings of Kähler Spaces Natural generalizations of geodesic mappings are the holomorphically projective mappings (HP-mappings) of Kähler spaces Kn . Naturally, similar problems appear within the HP-mappings theory as in the geodesic mappings theory. Interestingly, numerous findings and results valid for geodesic mappings seamlessly extend to HP-mappings as well, indicating a high degree of compatibility between the two. Note that HP-mappings were considered, as a rule, under the condition of the preservation of the structure. It turned out that in the case of HP-mappings, the structure is necessarily preserved. The works by Tashiro [ 63 ], Ishihara [ 50 ], Otsuki and Tashiro [ 21 ], Domashev and Mikeš [ 25 ], and Mikeš [ 6 , 7 , 26 , 64 , 65 ] are devoted to general questions concerning the theory of holomorphically projective mappings of the Kähler spaces Kn. Problems related to integrating the fundamental equations of HPM theory and other related questions have beenexamined, for example, intheworksbyAminovaand Kalinin [27–30] . Unfortunately, many of the questions that the authors present are not their own originally. The fundamentals of the theory of holomorphically projective mappings can be found in [ 22 ] by Beklemishev, [ 23 , 24 ] by Yano, [ 16 ] by Sinyukov, and [ 10 – 13 ] by Mikeš. In the monograph ([ 16 ], fifth chapter), Sinyukov presented classical results of holomorphically projective mappings, and results were obtained Mikeš and Domashev [25] and Mikeš [4,26]. Definitions and the Basic Equations Below, the terms related to holomorphically projective mappings and transformations are given in detail, e.g., [10–13,16,21–23]. An analytically planar curve γ of the Kähler space Kn is a curve defined by the equations x=x(t) , whose tangent vector λ=dγ(t)/dt , being translated, remains in the area Mathematics 2024,12, 1239 4 of 13 element formed by the tangent vector λ and its conjugate Fλ ; i.e., the conditions ∇tλ= ρ1(t)λ+ρ2(t)Fλ, where ρ1,ρ2are functions of the argument t, are fulfilled [21]. If ρ2(t)≡ 0, then ℓ is a geodesic. We note that if the tangent vector λ of an analytically planar curve γ is isotropic (null-vector) in one of its points, then it is isotropic in all its points γ , which is analogous to geodesics. The physical meaning of these curves is given, for example, in [66,67]. The diffeomorphism of Kn onto Kn is a holomorphically projective mapping if it transforms all the analytically planar curves of Knonto analytically planar curves of Kn. Under the HP-mapping, the structure of the spaces Kn and Kn is preserved; i.e., in the coordinate system x , in general, with respect to the mapping, the conditions Fh i(x) = Fh i(x) are satisfied. To be more precise, Fh i(x) = ±Fh i(x) for Kn , since the structure in Kn is defined with an accuracy within the sign (see [14]). The holomorphically projective mappings were introduced by Otsuki and Tashiro [ 21 ] for Kn under the a priori assumption that the structure was preserved. Note that HP-mappings are special F-planar mappings introduced by Mikeš and Sinyukov [ 38 ]. Questions about the preservation of the structure for the above mappings are studied in detail in [14,38,68]. The necessary and sufficient conditions for the holomorphically projective mappings of Kn onto Kn fulfill the following conditions in the general (with respect to the mapping) coordinate system (Tashiro [63]), Γh ij(x) = Γh ij(x) + ψiδh j+ψjδh i−ψıδh ȷ−ψȷδh ı, (1) where ψi is a vector, and Γh ij and Γh ij are the Christoffel symbols of Kn and Kn . Relation (1) is equivalent to the equation gij,k=2ψkgij +ψigjk +ψjgik +ψigjk +ψjgik, (2) where gij are the components of metric g on Kn . When ψi≡ 0, we say that the holomorphically projective mapping is nontrivial or affine. After contracting (1), it is valid that ψi is necessarily a gradient; moreover, ψi=∂iΨ, where Ψ=1 n+2ln s det g det g. The Riemannian and Ricci tensors Knand Knare connected by the conditions Rh ijk =Rh ijk +δh kψij −δh jψik +δh kψi j −δh jψi k +2δh iψj k;Rij =Rij −(n+2)ψij, where ψij ≡ψi,j−ψiψj+ψiψjis a symmetric tensor, for which ψij =ψi j. The holomorphically projective curvature tensor Ph ijk is invariant relative to the holomorphically projective mapping. Its identical vanishing is necessary and sufficient for Kn to be a space of constant holomorphic curvature and for these spaces to admit holomorphically projective mapping onto a flat space (Tashiro [ 63 ], Ishihara [ 50 ]). It has been proven that non-trivial holomorphically projective mapping can be established between any Kn of constant holomorphic curvature [14]. Mikeš [ 26 ] has found that the Kähler space Kn admits a holomorphically projective mapping if and only if the system of the following equations, (a) aij,k=λigjk +λjgik +λigjk +λjgik; (b) nλi,j=µgij +aiαRα j−aαβRα ·ijβ ·; (c) µ,i=2λαRα i, (3) has a solution for the unknown tensors aij (= aji =ai j , |aij| = 0 ) , λi and µ . The solutions of (2) and (3) are connected by the relations aij =e2ψgαβgαigβj , λi=−e2ψgαβgαiψβ . Evidently, λi=∂i( 2 aαβgαβ) is the gradient, and the mapping is trivial if and only if λi= 0. For vector Mathematics 2024,12, 1239 5 of 13 λi , it holds that λi,j=λi,j=λj,i ; therefore, λi,j+λj,i= 0. From this, it follows that the vector λiis the Killing vector. Condition (3)a is necessary and sufficient for the existence of the holomorphically projective mapping Kn; this result was obtained by Domashev and Mikeš [25]. Equation (3) forms a linear system of the Cauchy type with respect to the components of the unknown tensors aij , λi , and µ . Consequently, the general solution of this system depends on rhpm ≤(n/ 2 + 1 )2 parameters [ 25 ]. For rhpm > 2, Equations (4) and (5) hold; see [10–13,64] and [32]. The solution of Equation (3) in Kn reduces to the study of the integrability conditions for (3) and their differential continuations, which, in turn, constitute a system of linear algebraic equations for the unknows aij , λi , and µ . Thus, we can determine whether the given space Kn admits holomorphically projective mapping, and if it does, then with what arbitrariness. Holomorphically projective transformations of Kähler spaces are closely related to HP-mappings (see [ 50 , 51 , 59 , 69 , 70 ]). It is obvious that Kn , in which NHPT exist, admits NHPM, and conversely, there are no NHPT in the spaces Knthat do not admit NHPM. Mikeš [ 71 ] obtained the inequality rhpt ≤rhpm +r∗ m , where rhpt is the order of the complete group HPT, and r∗ m is the order of the complete group of motions that preserves the analytic planar curves. The spaces in which conditions hij,k=ψigjk +ψjgik +ψigjk +ψjgik are fulfilled necessarily admit HPT and, for B= 0, NHPT. A more detailed investigation of these regularities was carried out in [71]. Yamaguchi [ 72 ] studied a K -torse-forming vector ξ , for which ξh ,i=aδh i+bFh i+αiξh+ βiξαFh α . Esenov’s works [ 73 , 74 ] are devoted to the study of Kn in which there exist vector fields of this kind. He showed that K -torse-forming vector fields were HPM-invariant. In his works, he studied Kn in which the conditions λi,j=agij +c(λiλj−eλiλj) , where a , c are invariants, were satisfied. These spaces admit NHPM. The metric of the holomorphically projectively corresponding spaces Kn that contain K -concircular fields has been found in explicit form. These fields exist in spaces of constant holomorphic curvature. 4. Holomorphically Projective Mappings of the Spaces Kn[B] We denote the Kähler space Kn by Kn[B] if it admits a holomorphically projective mapping under which the relations (for details, see Mikeš’ dissertation [ 4 ], also, see [ 10 , 11 ]) (a) aij,k=λigjk +λjgik +λigjk +λjgik; (b) λi,j=µgij +B aij (4) are satisfied, where aij , (= aji =ai j , |aij| = 0 ) , λi(≡ 0 ) , µ , B are tensors, while B is uniquely determined by the space Kn . When B is a constant, then µ,i= 2 Bλi , and when B≡0, then µis a constant. Relations (4) are equivalent to relations (2), and ψij =B gij −B gij. (5) These conditions are fulfilled, in particular, under the holomorphically projective mappings of spaces of a constant holomorphic curvature [ 14 ] and for HP-mappings between Einstein spaces, while B=−R n(n+2) and B=−R n(n+2) , where R and R are scalar curvatures of Knand Kn, respectively. Spaces in which there are K -concircular fields are spaces Kn[B] . In the spaces Kn[ 0 ] and Kn[B] , B≡ const , fields of this kind necessarily exist. The spaces Kn[B] admit NHPM only on Kn[B] , with B and B being simultaneously constant or nonconstant. The spaces Kn[B] , B=const , admit holomorphically projective transformation (nontrivial for B= 0). Under the holomorphically projective mapping of Kn[B] onto Kn[B] , the tensors ∗ Zh ijk and ∗ Zij are invariant, where ∗ Zh ijk ≡Rh ijk −Bδh kgij −δh jgik +δh kgij −δh jgik +2δh igjk;∗ Zij ≡∗ Zα ijα. Mathematics 2024,12, 1239 6 of 13 The set of solutions of system (4) forms, for B=const , a special Jordan algebra relative to the multiplication operation (see [65,75]): 1 a,1 λ,1 µ×2 a,2 λ,2 µ=3 a,3 λ,3 µ, with 23 aij =B1 as ·(i 2 aj)s−1 λ(i 2 λj)−1 λ(i 2 λj); 2 3 λi=B(1 λα2 aiα+2 λα1 aiα)−1 µ2 λi−2 µ1 λi;3 µ=B1 λα2 λα−1 µ2 µ. A similar multiplication operation for solutions in Vn(B) was obtained by Mikeš and Shandra [76]. It was established [ 75 ] that every solution aij of (4) in Kn[B] , B=const = 0, is associated with a covariantly constant field Aab in the Riemannian space Vn+2 , whose metric tensor has the structure Gab =1 Be2Bx0  −B0 0 0gij −Bτiτj−Bτi 0−Bτj−B , where gij(x1 , . . . , xn) is the metric tensor of Kn[B] , B=const = 0, and τi(x1 , . . . , xn) is a covector potential; i.e., Fij =∂[jτi] (the form Fij ≡giαFα j is exact), a , b= 0,1, . . . , n , n+ 1, with Aab =   µ λi0 λjaij +τ(iFα j)λα+µτiτjλαFα i−µτi 0λαFα j−µτjµ  . Holomorphically Projective Mappings of T-Quasi-Semisymmetric Spaces The following terms and results, unless otherwise stated, were introduced in Mikeš’s dissertation [4] and publications [10–13]. By means of ∗ Zh ijk, we introduce into consideration the operation ⟨⟨lm⟩⟩ as follows: Th1...hp i1...iq⟨⟨lm⟩⟩ ≡ q ∑ s=1 Th1...hp i1...is−1αis+1...iq ∗ Zα islm −Th1...hs−1αhs+1...hp i1...iq ∗ Zhs αlm, where Tis a tensor of the type (p q). When B=0, T⟨⟨lm⟩⟩ =T,[lm]. For tensors uand v, this operation possesses the properties (u±v)⟨⟨lm⟩⟩ =u⟨⟨lm⟩⟩ ±v⟨⟨lm⟩⟩;(uv)⟨⟨lm⟩⟩ =u⟨⟨lm⟩⟩v+uv⟨⟨lm⟩⟩;gij⟨⟨lm⟩⟩ =0; gij ⟨⟨lm⟩⟩ =0; δi j⟨⟨lm⟩⟩ =0. The analog of the Walker identities [77] is valid: Rhijk⟨⟨lm⟩⟩ +Rjklm⟨⟨hi⟩⟩ +Rlmhi⟨⟨jk⟩⟩ =0. We say [ 4 , 8 , 10 ] that the Kähler space Kn is T -quasi-semisymmetric (TPs n [B]) if the condition T⟨⟨lm⟩⟩ = 0 is fulfilled in it. Many results regarding HP-mappings of these spaces can be found, for example, in [ 4 , 8 , 10 , 78 – 80 ]. Here, it was proved that HP-mappings of these spaces fulfill Equation (4) . Spaces for which Rh ijk⟨⟨lm⟩⟩ = 0 and Rij⟨⟨lm⟩⟩ = 0 (see [ 4 , 8 , 78 ]) were studied by Luczyszyn and Olszak, respectively [81–83]. In the works by Sinyukov, Sinyukova [ 55 ], and Mikeš [ 10 , 56 ], a series of results for global geodesic mappings of compact semisymmetric and Ricci-semisymmetric Kähler manifolds with additional conditions was obtained. Haddad proved that the four-dimensional Einsteinian Kn spaces do not admit NHPM onto the Einsteinian spaces of nonconstant holomorphic curvature and do not admit nontrivial holomorphically projective transforma- Mathematics 2024,12, 1239 7 of 13 tions. The investigation of the NHPM of complete Einsteinian Kn was carried out in [ 54 ] by Akbar-Zadeh. 5. Rigidity of the Kähler Spaces’ Respective Holomorphically Projective Mappings 5.1. Spaces That Do Not Admit Nontrivial HPM Locally Many authors isolated Kähler spaces that do not admit either nontrivial holomorphically projective mappings (NHPM) or nontrivial holomorphically projective transformations (NHPT). Note that the Kähler spaces Kn , which do not admit NHPM, do not admit NHPT either, as well as nontrivial geodesic mappings or nontrivial projective transformations. In these spaces, there are no nonconstant concircular and K -concircular vector fields. In this section, this is not specifically stipulated. In 1974, Sakaguchi [ 61 ] proved that proper Kähler symmetric spaces Kn of nonconstant holomorphic curvature do not admit NHPM. For symmetric Kn with a metric of arbitrary signature, Sakaguchi’s result was proved by Domashev and Mikeš [ 25 ] (see also [ 10 – 13 , 16 ]). In [ 8 , 10 ], Mikeš indicated more general conditions for recurence under which Kn does not admit NHPM. In particular, recurrent, m -recurrent, two-symmetric, and generalized recurrent D2 nKähler spaces K± ndo not admit NHPM. The abovementioned results for holomorphically projective mappings of semisymmetric and generalized recurrent manifolds with affine connection were generalized in papers [10,84–86] by al Lamy, Mikeš, Škodová, etc. 5.2. Holomorphically Complete Manifolds Kn[B] I. Hasegawa and K. Yamauchi in [ 52 ] proved that an infinitesimal holomorphically projective transformation has infinitesimal isometry on a compact classic Kähler manifold Kn with non-positive constant scalar curvatur. Additionally, they proved that a compact classical Kähler manifold with constant scalar curvature is holomorphically isometric to a complex projective space with the Fubini–Study metric (i.e., manifold with constant holomorphic curvature), provided Kn admits a non-isometric infinitesimal holomorphically projective transformation. The investigation of the holomorphically projective mappings of the complete Einstein Kähler manifold Knwas carried out by H. Akbar-Zadeh and R. Couty in [53,54,87]. We prove the following theorem ([13], p. 502). Theorem 1. Let a Kähler manifold Kn[B] , B=const , admit a holomorphically projective mapping f onto a complete manifold Kn. 1. If Kn[B]has an indefinite metric, then f is affine. 2. If B ≥0, then f is affine. Please note that the proof presented there is not correct. Below, we prove more general facts from which this Theorem follows. 5.3. Holomorphically Projective Mappings and Fundamental Functions along Geodesics Let us suppose that f : Kn→Kn is a holomorphically projective mapping and Equation (4) holds with ψi=∂iΨ ; B and B are constants. Let γ(s) be a geodesic on Kn and a corresponding analytically planar curve γ(τ(s)) on Kn with natural parameter s and with canonical parameter τ , respectively. Assume ˙ τ=dτ(s)/ds > 0 for the parameter transformation τ=τ(s). Because gand e−4Ψgare first integrals of geodesics, the following holds: gij ˙ γi˙ γj=ε=±1,0 and gij ˙ γi˙ γj=c e4Ψ(t),c=const . (6) The first equality is generally known, and the second follows from the contraction of (2) with ˙ γi˙ γj˙ γk. Mathematics 2024,12, 1239 8 of 13 By differentiating γ(s) = γ(τ(s)) with respect to parameter s, we obtain ˙ γ(s) = ◦ γ(τ(s)) ·˙ τ(s), where ◦ γ=dγ(τ) dτ, (7) and, naturally, we suppose that ˙ τ(s)>0. Since τ is canonical of γ it holds that g(◦ γ , ◦ γ) = c(= const ) , and from this, it follows that g(˙ γ , ˙ γ) = c·˙ τ2 . Then, from (7) , in the case where c= 0 (and c= 0), the following holds: ˙ τ(s) = e c·e2Ψ,e c>0; (8) i.e., γ is a non-isotropic analytical planar curve on Kn . In the case where c= 0 (and c= 0), this formula may not apply. Along the geodesic γ(s) , we put Ψ(s) = Ψ(γ(s)) , and from this, ˙ Ψ(s) = ψα˙ γα . For the tensor ψij (≡ψi,j−ψiψj+ψiψj) , the equality ψij =ψji =ψi j holds. It follows from this that tensor ψi j is skew, and ψα β ˙ γα˙ γβ=0. From the definition of ψij, evidently, ψi,j=ψiψj+ψiψj+ψi j . By differentiating the expression ψα˙ γα with respect to s , from the above, we make sure that (ψα˙ γα).=2˙ Ψ·(ψα˙ γα), and, after integrating, it is obvious that ψα˙ γα=χ·e2Ψ, where χis constant. Next, we study the holomorphically projective mapping where the condition (4) is valid, i.e., ψij =B gij −B gij , where B and B are constants. If this mapping is non-trivial, then the spaces Knand Knwill be Kn[B]and Kn[B], respectively. We can write the condition (4) in expanded form ψi,j=ψiψj−ψiψj+B gij −B gij . (9) We calculate ¨ Ψ(s) according to geodesics γ(s) : ¨ Ψ(s) = ( ˙ Ψ).= (ψα˙ γα).=ψα,β˙ γα˙ γβ , and after using (9), we obtain ¨ Ψ= ( ˙ Ψ)2+b·e4Ψ−a, (10) where a=εB, and b=c B −χ2. We substitute q=e−2Ψ(s); then, Equation (9) is equivalent to 2q¨ q=˙ q2−4b+4a q2. (11) The derivative of (11) gives the following equation ... q=4a˙ q, which has a solution (a)q=c0+c1s+c2s2, if a=0, (b)q=c0+c1cosh(αs) + c2sinh(αs), if a>0, (c)q=c0+c1cos(αs) + c2sin(αs), if a<0, (12) where α= 2 p|a| , and c0 , c1 , c2 are constants. Since the function q must satisfy Equation (11), the coefficients ciare tied to each other. We analyze the obtained results in terms of the compactness and completeness of the studied geodesics γand their image γ=f(γ). Lemma 1. If geodesic γ on Kn is compact, then, for a≡B·g(˙ γ , ˙ γ)≥ 0, the function Ψ(s) is constant. Mathematics 2024,12, 1239 9 of 13 Lemma 2. If geodesic γ on Kn and its non-isotropic images γ on Kn are complete, then, for a≡B·g(˙ γ,˙ γ)≥0, the function Ψ(s)is constant. Lemma 3. If geodesic γ on Kn is complete, then, for a≡B·g(˙ γ , ˙ γ) = 0and b≡B c −χ2= 0, the function Ψ(s)is constant. Proof. The proof of Lemma 1is trivial, because the non-constant function Ψ(s) is not bounded for s∈R. The proof of the analogue of Lemma 2has been shown by many authors and relies on ideas by Couty [ 87 ] in an investigation of projective transformations of Einstein manifolds and by Shen [88] in an investigation of Finsler Einstein geodesically equivalent metrics. Since q=e−2Ψ , from (8), it follows that ˙ τ(s) = e c e2Ψ=e c/q(s)> 0. We mean τ(s)=Rs s0e c/q(t)dt. For functions q=c0+c1s+c2s2 and q=c0+c1cosh(αs) + c2sinh(αs) , (c1= 0 or c2= 0 ) , this integral diverges (goes to infinity in finite time s ). Then, c1=c2= 0 and τ=const ·s+s0 , and it follows that ˙ τ=const . Evidently, the function Ψ(s) is constant along geodesic γ(s). The proof of Lemma 3is trivial, because for non-constant function q(s) , there exists s0 , for which q(s0) = 0; this is the contradiction with q(s)>0 for s∈R. 5.4. Holomorphically Projective Mappings of Kn[0] with n Complete Geodesics Holomorphically projective mapping Kn [0] onto Kn[B] is characterized by Equation (2) and ψij ≡ψi,j−ψiψj+ψiψj=B gij , (13) which are equivalent to the equations (see Formulae (4)a and (5)) λi,j=µgij ,µ=const . (14) Theorem 2. Let g be a (pseudo-) Riemannian Kähler metric, its complex structure F on a domain V of n-dimensional manifold M ( shortly Kähler space Kn = (M , g , F)) , and their holomorphically projective mapping of Kn onto Kähler space Kn with Equation (13). Further, assume that there is a point at which not all sectional curvatures are vanishing and through which in linearly independent directions pass n/2 complete geodesics, for which the condition of at least one of Lemmas 1–3applies. These directions with their complex united vectors form an n -dimension base. Then, this mapping is trivial (affine). Proof. Let the conditions of the theorem be satisfied. Then, according to the given geodesics, the function Ψ(s) is constant; thus, at point x0 in the direction of these geodesics, ∂αΨ(x)˙ γα= 0 is vanishing in the tangent directions. Since this also applies to complex united directions at point x0 : ∂αΨ(x)˙ γα= 0, then ψi(x0) = 0 must apply. This is equivalent with λi(x0) = 0. The integrability conditions of Equation (14) have the form λαRα ijk =0. We covariantly differentiate them and use (14): µRlijk +λαRα ijk,l= 0. It follows that µ= 0 to the extent that Rhijk(x0)= 0. Therefore, equations λi,j= 0 for initial conditions λi(x0) = 0 have a trivial solution λi(x) = 0. It follows that ψi(x) is vanishing, and holomorphically projective mapping is trivial (in other words affine). Note that, in the above assumption, there do not exist Kn [0] and Kn [ B ], which are holomorphically projective correspondent. 5.5. Holomorphically Projective Mappings of Kn[B] with Finite Complete Geodesics For spaces Kn [B], B= 0 the similar condition is weak. Therefore, we recall some aspects of matrix theory.