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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**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]).

## 2. Kähler Spaces

## 3. General Questions Concerning Holomorphically Projective Mappings of Kähler Spaces

#### Definitions and the Basic Equations

## 4. Holomorphically Projective Mappings of the Spaces ${\mathit{K}}_{\mathit{n}}\left[\mathit{B}\right]$

#### Holomorphically Projective Mappings of T-Quasi-Semisymmetric Spaces

_{n}[B]) if the condition ${T}_{\langle \phantom{\rule{-0.166667em}{0ex}}\langle lm\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle}=0$ is fulfilled in it. Many results regarding HP-mappings of these spaces can be found, for example, in [4,8,10,78,79,80]. Here, it was proved that HP-mappings of these spaces fulfill Equation (4). Spaces for which ${R}_{ijk\langle \phantom{\rule{-0.166667em}{0ex}}\langle lm\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle}^{h}=0$ and ${R}_{ij\langle \phantom{\rule{-0.166667em}{0ex}}\langle lm\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle}=0$ (see [4,8,78]) were studied by Luczyszyn and Olszak, respectively [81,82,83].

## 5. Rigidity of the Kähler Spaces’ Respective Holomorphically Projective Mappings

#### 5.1. Spaces That Do Not Admit Nontrivial HPM Locally

**Note that the Kähler spaces**${K}_{n}$, 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.

#### 5.2. Holomorphically Complete Manifolds ${K}_{n}\left[B\right]$

**Theorem**

**1.**

#### 5.3. Holomorphically Projective Mappings and Fundamental Functions along Geodesics

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

#### 5.4. Holomorphically Projective Mappings of ${K}_{n}$[0] with n Complete Geodesics

**Theorem**

**2.**

**Proof.**

#### 5.5. Holomorphically Projective Mappings of ${K}_{n}$[B] with Finite Complete Geodesics

**Theorem**

**3.**

**through which in directions**${v}_{i}\in {S}^{2\ast}$ pass ${(n/2)}^{2}$ complete geodesics, for which the condition of at least one of Lemmas 1–3 applies. Then, this mapping is homothetic; i.e., the metrics are proportional with a constant coefficient.

**Proof.**

## 6. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Westlake, W.J. Hermitian spaces in geodesic correspondence. Proc. AMS
**1954**, 5, 301–303. [Google Scholar] [CrossRef] - Yano, K.; Nagano, T. Some theorems on projective and conformal transformations. Koninkl. Nederl. Akad. Wet.
**1957**, 60, 451–458. [Google Scholar] [CrossRef] - Muto, Y. On some special Kählerian spaces. Sci. Rep. Yokogama Natl. Univ.
**1961**, 1, 1–8. [Google Scholar] - Mikeš, J. Geodesic and Holomorphically Projective Mappings of Special Riemannian Space. Ph.D. Thesis, Odessa University, Odessa, Ukraine, 1979. [Google Scholar]
- Mikeš, J. On geodesic mappings of 2-Ricci symmetric Riemannian spaces. Math. Notes
**1981**, 28, 622–624. [Google Scholar] [CrossRef] - Mikeš, J. Equidistant Kähler spaces. Math. Notes
**1985**, 38, 855–858. [Google Scholar] [CrossRef] - Mikeš, J. On Sasaki spaces and equidistant Kähler spaces. Sov. Math. Dokl.
**1987**, 34, 428–431. [Google Scholar] - Mikeš, J. Geodesic, F-Planar and Holomorphically Projective Mappings of Riemannian Spaces and Spaces with Affine Connections. Ph.D. Thesis, Charles University, Prague, Czech Republic, 1996. [Google Scholar]
- Mikeš, J. Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. N. Y.
**1996**, 78, 311–333. [Google Scholar] [CrossRef] - Mikeš, J. Holomorphically projective mappings and their generalizations. J. Math. Sci. N. Y.
**1998**, 89, 1334–1353. [Google Scholar] [CrossRef] - Mikeš, J.; Vanžurová, A.; Hinterleitner, I. Geodesic Mappings and Some Generalizations; Palacký University Olomouc: Olomouc, Czech Republic, 2009. [Google Scholar]
- Mikeš, J.; Stepanova, E.; Vanžurová, A.; Bácsó, S.; Berezovski, V.E.; Chepurna, O.; Chodorová, M.; Chudá, H.; Gavrilchenko, M.L.; Haddad, M. Differential Geometry of Special Mappings; Palacký University Olomouc: Olomouc, Czech Republic, 2015; 566p, ISBN 978-80-244-4671-4/pbk. [Google Scholar]
- Mikeš, J.; Stepanova, E.; Vanžurová, A.; Bácsó, S.; Berezovski, V.E.; Chepurna, O.; Chodorová, M.; Chudá, H.; Gavrilchenko, M.L.; Haddad, M. Differential Geometry of Special Mappings, 2nd ed.; Palacký University Olomouc: Olomouc, Czech Republic, 2019; 674p, ISBN 978-80-244-5535-8/pbk. [Google Scholar]
- Sinyukov, N.S.; Kurbatova, I.N.; Mikeš, J. Holomorphically Projective Mappings of Kähler Spaces; Odessa University Press: Odessa, Ukraine, 1985. [Google Scholar]
- Sinyukov, N.S. On geodesic mappings of Riemannian manifolds onto symmetric spaces. Dokl. Akad. Nauk SSSR
**1954**, 98, 21–23. [Google Scholar] - Sinyukov, N.S. Geodesic Mappings of Riemannian Spaces; Nauka: Moscow, Russia, 1979. [Google Scholar]
- Shirokov, P.A. Constant fields of vectors and tensors of second order on Riemannian spaces. Kazan Učen. Zap. Univ.
**1925**, 25, 256–280. [Google Scholar] - Shirokov, P.A. Selected Investigations on Geometry; Kazan University Press: Kazan, Russia, 1966. [Google Scholar]
- Shirokov, A.P. Shirokov’s work on the geometry of symmetric spaces. J. Math. Sci.
**1998**, 89, 1253–1260. [Google Scholar] [CrossRef] - Yano, K. Concircular Geometry. Proc. Imp. Acad. Tokyo
**1940**, 16, 195–200, 354–360, 442–448, 505–511. [Google Scholar] - Otsuki, T.; Tashiro, Y. On curves in Kaehlerian spaces. Math. J. Okayama Univ.
**1954**, 4, 57–78. [Google Scholar] - Beklemišev, D.V. Differential geometry of spaces with almost complex structure. Geom. Itogi Nauki Tekhn.
**1965**, 2, 165–212. [Google Scholar] - Yano, K. Differential Geometry of Complex and Almost Comlex Spaces; Pergamon Press: Oxford, UK, 1965. [Google Scholar]
- Yano, K.; Bochner, S. Curvature and Betti Numbers; Princeton University Press: Princeton, NJ, USA, 1953. [Google Scholar]
- Domashev, V.V.; Mikeš, J. Theory of holomorphically projective mappings of Kählerian spaces. Math. Notes
**1978**, 23, 160–163. [Google Scholar] [CrossRef] - Mikeš, J. On holomorphically projective mappings of Kählerian spaces. Ukr. Geom. Sb.
**1980**, 23, 90–98. [Google Scholar] - Aminova, A.V.; Kalinin, D.A. H-projectively equivalent four-dimensional Riemannian connections. Russ. Math.
**1994**, 38, 10–19. [Google Scholar] - Aminova, A.V.; Kalinin, D.A. Quantization of Kähler manifolds admitting H-projective mappings. Tensor New Ser.
**1995**, 56, 1–11. [Google Scholar] - Aminova, A.V.; Kalinin, D.A. H-projective mappings of four-dimensional Kähler manifolds. Russ. Math.
**1998**, 42, 1–11. [Google Scholar] - Aminova, A.V.; Kalinin, D.A. Lie algebras of H-projective motions of Kähler manifolds of constant holomorphic sectional curvature. Math. Notes
**1999**, 65, 679–683. [Google Scholar] [CrossRef] - Ishihara, T. Some integral formulas in Fubini-Study spaces. J. Math. Tokushima Univ.
**1985**, 19, 19–23. [Google Scholar] - Fedorova, A.; Kiosak, V.; Matveev, V.S.; Rosemann, S. The only Kähler manifold with degree of mobility at least 3 is (CP(n),g
_{Fubini-Study}). Proc. Lond. Math. Soc.**2012**, 105, 153–188. [Google Scholar] [CrossRef] - Prvanović, M. Holomorphically projective transformations in a locally product space. Math. Balk.
**1971**, 1, 195–213. [Google Scholar] - Kurbatova, I.N. HP-mappings of H-spaces. Ukr. Geom. Sb.
**1984**, 27, 75–83. [Google Scholar] - Peška, P.; Mikeš, J.; Chudá, H.; Shiha, M. On holomorphically projective mappings of parabolic Kähler manifolds. Miskolc Math. Notes
**2016**, 17, 1011–1019. [Google Scholar] [CrossRef] - Petrov, A.Z. Modeling of physical fields. Gravit. Gen. Relat.
**1968**, 4, 7–21. [Google Scholar] - Bejan, C.-L.; Kowalski, O. On generalization of geodesic and magnetic curves. Note Mat.
**2017**, 37, 49–57. [Google Scholar] - Mikeš, J.; Sinyukov, N.S. On quasiplanar mappings of spaces of affine connection. Sov. Math.
**1983**, 27, 63–70. [Google Scholar] - Vesić, N.O.; Velimirović, L.S.; Stanković, M.S. Some invariants of equitorsion third type almost geodesic mappings. Mediterr. J. Math.
**2016**, 13, 4581–4590. [Google Scholar] [CrossRef] - Kozak, A.; Borowiec, A. Palatini frames in scalar-tensor theories of gravity. Eur. Phys. J.
**2019**, 79, 335. [Google Scholar] [CrossRef] - Vishnevsky, V.V. Affinor structures of manifolds as structures defined by algebras. Tensor
**1972**, 26, 363–372. [Google Scholar] - Vishnevsky, V.V.; Shirokov, A.P.; Shurigin, V.V. Spaces ver Algebras; Kazan Univerisity Press: Kazan, Russia, 1985. (In Russian) [Google Scholar]
- Evtushik, L.E.; Lumiste, Y.G.; Ostianu, N.M.; Shirokov, A.P. Differential-geometric structures on manifolds. J. Sov. Math.
**1980**, 14, 1573–1719. [Google Scholar] [CrossRef] - Kobayashi, S.; Nomizu, K. Foundations of Differential Geometry; Interscience Publishers Inc.: New York, NY, USA, 1963; Volume 1. [Google Scholar]
- Kobayashi, S.; Nomizu, K. Foundations of Differential Geometry; Interscience Publishers Inc.: New York, NY, USA, 1969; Volume 2. [Google Scholar]
- Kobayashi, S. Transformation Groups in Differential Geometry; Springer: Berlin/Heidelberg, Germany, 1972. [Google Scholar]
- Hall, G. Four-Dimensional Manifolds and Projective Structure; CRC Press: Boca Raton, FL, USA, 2023; ISBN 9780367900427. [Google Scholar]
- Deszcz, R.; Hotloś, M. Notes on pseudo-symmetric manifolds admitting special geodesic mappings. Soochow J. Math.
**1989**, 15, 19–27. [Google Scholar] - Stepanov, S.E. New methods of the Bochner technique and their applications. J. Math. Sci.
**2003**, 113, 514–536. [Google Scholar] [CrossRef] - Ishihara, S. Holomorphically projective changes and their groups in an almost complex manifold. Tohoku Math. J. II Ser.
**1957**, 9, 273–297. [Google Scholar] [CrossRef] - Tachibana, S.-I.; Ishihara, S. On infinitesimal holomorphically projective transformations in Kählerian manifolds. Tohoku Math. J. II Ser.
**1960**, 12, 77–101. [Google Scholar] [CrossRef] - Hasegawa, I.; Yamauchi, K. On infinitesimal holomorphically projective transformations in compact Kaehlerian manifolds. Hokkaido Math. J.
**1979**, 8, 214–219. [Google Scholar] [CrossRef] - Akbar-Zadeh, H.; Couty, R. Espaces a tenseur de Ricci parallele admetant des transformations projectives. Rend. Mat.
**1978**, 11, 85–96. [Google Scholar] - Akbar-Zadeh, H. Sur les transformations holomorphiquement projectives de varietes Hermitiannes et Kähleriannes. C.R. Acad. Sci.
**1987**, 304, 335–338. [Google Scholar] - Sinyukov, N.S.; Sinyukova, E.N. Holomorphically projective mappings of special Kähler spaces. Math. Notes
**1984**, 36, 706–709. [Google Scholar] [CrossRef] - Mikeš, J. On Global Concircular Vector Fields on Compact Riemannian Spaces; No. 615-Uk88; State Archival Service of Ukraine: Kyiv, Ukraine, 1988. [Google Scholar]
- Afwat, M.; Švec, A. Global differential geometry of hypersurfaces. Rozpr. ČSAV
**1978**, 88, 75. [Google Scholar] - Mikeš, J. Global geodesic mappings and their generalizations for compact Riemannian spaces. Silesian Univ. Math. Publ.
**1993**, 1, 143–149. [Google Scholar] - Tachibana, S.-I.; Ishihara, S. A note on holomorphically projective transformations of a Kählerian space with parallel Ricci tensor. Tohoku Math. J. II Ser.
**1961**, 13, 193–200. [Google Scholar] - Bácsó, S.; Ilosvay, F. On holomorphically projective mappings of special Kaehler spaces. Acta Math. Acad. Paedagog. Nyházi.
**1999**, 15, 41–44. [Google Scholar] - Sakaguchi, T. On the holomorphically projective correspondence between Kählerian spaces preserwing complex structure. Hokkaido Math. J.
**1974**, 3, 203–212. [Google Scholar] [CrossRef] - Kähler, E. Über eine bemerkenswerte Hermitische Metric. Abh. Math. Semin. Hamburg. Univ.
**1933**, 9, 173–186. [Google Scholar] [CrossRef] - Tashiro, Y. On holomorphically projective correspondence in an almost complex space. Math. J. Okayama Univ.
**1957**, 6, 147–152. [Google Scholar] - Mikeš, J. F-planar mappings of spaces of affine connection. Arch. Math. Brno
**1991**, 27a, 53–56. [Google Scholar] - Mikeš, J. On holomorphically projective mappings of Kähler spaces. In Proceedings of the Conference Dedicated to the 200th Anniversary of N.I. Lobachevsky, Odessa, Ukraine, 3–8 September 1992; Abstract Report Part I. Odessa State University Press: Odessa, Ukraine, 1992; p. 80. (In Russian). [Google Scholar]
- Kalinin, D.A. Trajectories of charged particles in Kähler magnetic fields. Rep. Math. Phys.
**1997**, 39, 299–309. [Google Scholar] [CrossRef] - Kalinin, D.A. H-projectively equivalent Kähler manifolds and gravitational instantons. Nihonkai Math. J.
**1998**, 9, 127–142. [Google Scholar] - Hinterleitner, I.; Mikeš, J.; Peška, P. Fundamental equations of F-planar mappings. Lobachevskii J. Math.
**2017**, 38, 653–659. [Google Scholar] [CrossRef] - Mizusawa, H. On infinitesimal holomorphically projective transformations in
^{*}O-spaces. Tohoku Math. J. II Ser.**1961**, 13, 466–480. [Google Scholar] [CrossRef] - Kashiwada, T. Notes on infinitesimal HP-transformations in Kähler manifolds with constant scalar curvature. Natur. Sci. Rep. Ochanomizu Univ.
**1974**, 25, 67–68. [Google Scholar] - Mikeš, J. On an order of special transformation of Riemannian spaces. In Differential Geometry and Its Applications, Proceedings of the Conference, Dubrovnik, Yugoslavia, June 26–July 3, 198; Faculty of Mathematics, University of Belgrade: Belgrade, Serbia, 1989; pp. 199–208. [Google Scholar]
- Yamaguchi, S. On Kählerian torse-forming vector fields. Kodai Math. J.
**1979**, 2, 103–115. [Google Scholar] [CrossRef] - Esenov, K.R. On Generalized Geodesic and Geodesic Mappings of Special Riemannian Spaces. Ph.D. Thesis, Kyrgyz National University, Bishkek, Kyrgyzstan, 1993. (Supervisors Mikeš, J. and Borubayev, A.). [Google Scholar]
- Esenov, K.R. On properties of generalized equidistant Kählerian spaces admitting special, almost geodesic mappings of the second type. Collect. Sci. Works Frunze. 1988, pp. 81–84. Available online: https://zbmath.org/0732.53054 (accessed on 30 March 2024).
- Shandra, I.G. Geodesic mappings of equidistant spaces and Jordan algebras of spaces V
_{n}(K). Diff. Geom. Mnogoobr. Fig.**1993**, 24, 104–111. [Google Scholar] - Shandra, I.G.; Mikeš, J. Geodesic mappings of V
_{n}(K)-spaces and concircular vector fields. Mathematics**2019**, 7, 692. [Google Scholar] [CrossRef] - Walker, A.G. On Ruse’s spaces of recurrent curvature. Proc. London Math. Soc.
**1950**, 2, 36–64. [Google Scholar] [CrossRef] - Mikeš, J.; Radulović, Ž.; Haddad, M. Geodesic and holomorphically projective mappings of m-pseudo- and m-quasisymmetric Riemannian spaces. Russ. Math.
**1996**, 40, 28–32. [Google Scholar] - Haddad, M. Holomorphically Projective Mappings of Kählerian Spaces. Ph.D. Thesis, Moscow State University, Moscow, Russia, 1995. (Supervisors Evtushik, L.E. and Mikeš, J.). [Google Scholar]
- Haddad, M. Holomorphically-projective mappings of T-quasisemisymmetric and generally symmetric Kählerian spaces. DGA Silesian Univ. Math. Publ.
**1993**, 1, 137–141. [Google Scholar] - Luczyszyn, D. On pseudosymmetric para-Kählerian manifolds. Beitr. Algebra Geom.
**2003**, 44, 551–558. [Google Scholar] - Luczyszyn, D.; Olszak, Z. On paraholomorphically pseudosymmetric para-Kählerian manifolds. J. Korean Math. Soc.
**2008**, 45, 953–963. [Google Scholar] [CrossRef] - Olszak, Z. On compact holomorphically pseudosymmetric Kählerian manifolds. Cent. Eur. J. Math.
**2009**, 7, 442–451. [Google Scholar] [CrossRef] - al Lamy Raad, J.; Škodová, M.; Mikeš, J. On holomorphically projective mappings from equiaffine generally recurrent spaces onto Kählerian spaces. Arch. Math.
**2006**, 42, 291–299. [Google Scholar] - Mikeš, J.; Škodová, M.; al Lamy, R.J. On holomorphically projective mappings from equiaffine special semisymmetric spaces. In Proceedings of the 5th International Conference Aplimat, II, Bratislava, Slovak Republic, 7–10 February 2006; pp. 113–121. [Google Scholar]
- Škodová, M.; Mikeš, J.; Pokorná, O. On holomorphically projective mappings from equiaffine symmetric and recurrent spaces onto Kählerian spaces. Rend. Circ. Matem. Palermo. Ser. II Suppl.
**2005**, 75, 309–316. [Google Scholar] - Couty, R. Transformations infinitésimales projectives. C. R. Acad. Sci.
**1958**, 247, 804–806. [Google Scholar] - Shen, Z. On projectively related Einstein metrics in Riemann-Finsler geometry. Math. Ann.
**2001**, 1, 625–647. [Google Scholar] [CrossRef]

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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

**AMA Style**

Vítková L, Hinterleitner I, Mikeš J.
Rigidity of Holomorphically Projective Mappings of Kähler Spaces with Finite Complete Geodesics. *Mathematics*. 2024; 12(8):1239.
https://doi.org/10.3390/math12081239

**Chicago/Turabian Style**

Vítková, Lenka, Irena Hinterleitner, and Josef Mikeš.
2024. "Rigidity of Holomorphically Projective Mappings of Kähler Spaces with Finite Complete Geodesics" *Mathematics* 12, no. 8: 1239.
https://doi.org/10.3390/math12081239