On geodesic definiteness by similarity points

I. Hinterleitner ; N. I. Guseva ; J. Mikeš

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 4, Tome 182 (2020), pp. 19-27.

Résumé

In this paper, we present some results obtained in the theory of geodesic mappings of surfaces. It is well known that a mapping that is both conformal and geodesic is homothetic. Based on this property, we obtain new results on the definiteness of surfaces with respect to geodesic mappings, which generalize results obtained by V. T. Fomenko.

Détail

Keywords: geodesic mapping, conformal mapping, pseudo-Riemannian space, Riemannian space, definiteness.
Mots-clés : surface

Bibliographie

[1] Berezovskii V. E., Guseva N. I., Mikesh I., “O chastnom sluchae pochti geodezicheskikh otobrazhenii pervogo tipa prostranstv affinnoi svyaznosti, pri kotorom sokhranyaetsya nekotoryi tenzor”, Mat. zametki., 98:3 (2015), 463–466 | MR | Zbl

[2] Berezovskii V. E., Ginterleitner I., Guseva N. I., Mikesh I., “Konformnye otobrazheniya rimanovykh prostranstv na Richchi simmetricheskie prostranstva”, Mat. zametki., 103:2 (2018), 303–306 | MR | Zbl

[3] Evtushik L. E., Ginterleitner I., Guseva N. I., Mikesh I., “Konformnye otobrazheniya na prostranstva Einshteina”, Izv. vuzov. Mat., 2016, no. 10, 8–13 | MR | Zbl

[4] Kagan V. F., Osnovy teorii poverkhnostei. T. 1, 2, OGIZ, M.-L., 1947–1948

[5] Mikesh I., “O suschestvovanii $n$-mernykh kompaktnykh prostranstv, dopuskayuschikh netrivialnykh geodezicheskikh otobrazhenii «v tselom»”, Dokl. AN SSSR., 305 (1989), 534–536 | MR | Zbl

[6] Mikesh I., Ginterleitner I., “Ob odnom variatsionnom svoistve geodezicheskikh v rimanovykh i finslerovykh prostranstvakh”, Vestn. Tver. un-ta. Ser. Prikl. mat., 8 (2008), 59–63

[7] Norden A. P., Teoriya poverkhnostei, GITTL, M., 1956

[8] Petrov A. Z., Novye metody v obschei teorii otnositelnosti, Nauka, M., 1966

[9] Sinyukov N. S., “Ob odnom invariantnom preobrazovanii rimanovykh prostranstv s obschimi geodezicheskimi”, Dokl. AN SSSR., 137:6 (1961), 1312–1314 | MR | Zbl

[10] Sinyukov N. S., Geodezicheskie otobrazheniya rimanovykh prostranstv, Nauka, M., 1979 | MR

[11] Sinyukova E. N., “O geodezicheskikh otobrazheniyakh nekotorykh spetsialnykh rimanovykh prostranstv”, Mat. zametki., 30:6 (1981), 889–894 | MR | Zbl

[12] Stepanov S. E., “K globalnoi teorii proektivnykh otobrazhenii”, Mat. zametki., 58:1 (1995), 111–118 | MR | Zbl

[13] Shirokov P. A., Izbrannye raboty po geometrii, Izd. Kazansk. un-ta, Kazan, 1966 | MR

[14] Fomenko V. T., “Ob odnoznachnoi opredelennosti zamknutykh poverkhnostei otnositelno geodezicheskikh otobrazhenii”, Dokl. RAN., 407:4 (2006), 453–456 | MR | Zbl

[15] Afwat M., Švec A., “Global differential geometry of hypersurfaces”, Rozpr. Česk. Akad. Věd, Řada Mat. Pří{r}. Věd., 88:7 (1978), 75 | MR | Zbl

[16] Chudá H., Mikeš J., “First quadratic integral of geodesics with certain initial conditions”, APLIMAT, 2007, 85–88

[17] Chudá H., Mikeš J., “Conformally geodesic mappings satisfying a certain initial condition”, Arch. Math. (Brno)., 47 (2011), 389–394 | MR | Zbl

[18] Ćirić M., Zlatanović M., Stanković M., Velimirović Lj., “On geodesic mappings of equidistant generalized Riemannian spaces”, Appl. Math. Comput., 218 (2012), 6648–6655 | DOI | MR | Zbl

[19] Dini U., “On a problem in the general theory of the geographical representations of a surface on another”, Anali di Mat., 3 (1869), 269–294 | DOI

[20] Eisenhart L. P., Riemannian Geometry, Princeton Univ. Press, 1926 | MR | Zbl

[21] Formella S., Mikeš J., “Geodesic mappings of Einstein spaces”, Ann. Sci. Stet., 9 (1994), 31–40 | MR

[22] Hall G., “Projective structure in space-times. Adv. in Lorentzian Geometry”, AMS/IP Stud. Adv. Math., 49 (2011), 71–79 | DOI | MR | Zbl

[23] Hinterleitner I., “Conformally-projective harmonic diffeomorphisms of equidistant manifolds”, Publ. de la RSME., 11 (2007), 298–303 | MR | Zbl

[24] Hinterleitner I., Mikeš J., “On the equations of conformally-projective harmonic mappings”, AIP Conf. Proc., 956 (2007), 141–148 | DOI | MR | Zbl

[25] Hinterleitner I., “Special mappings of equidistant spaces”, J. Appl. Math., 2 (2008), 31–36

[26] Hinterleitner I., Mikeš J., “Projective equivalence and spaces with equi-affine connection”, J. Math. Sci., 177 (2011), 546–550 | DOI | MR | Zbl

[27] Hinterleiner I., Mikeš J., “Fundamental equations of geodesic mappings and their generalizations”, J. Math. Sci., 174 (2011), 537–554 | DOI | MR

[28] Hinterleitner I., “On global geodesic mappings of ellipsoids”, AIP Conf. Proc., 1460 (2012), 180–184 | DOI

[29] Hinterleitner I., “Geodesic mappings on compact Riemannian manifolds with conditions on sectional curvature”, Publ. Inst. Math., 94 (108) (2013), 125–130 | DOI | MR | Zbl

[30] Hinterleiner I., Mikeš J., “Geodesic mappings of (pseudo-) Riemannian manifolds preserve class of differentiability”, Miskolc Math. Notes., 14 (2013), 89–96 | DOI | MR

[31] Hinterleitner I., Mikeš J., “Geodesic mappings and Einstein spaces”, Geometric Methods in Physics, Birkhäuser–Springer, Basel, 2013, 331–335 | DOI | MR | Zbl

[32] Hinterleitner I., Mikeš J., “Geodesic mappings and differentiability of metrics, affine and projective connections”, Filomat., 29 (2015), 1245–1249 | DOI | MR | Zbl

[33] Hinterleiner I., Mikeš J., “Geodesic mappings onto Riemannian manifolds and differentiability”, Geometry, Integrability and Quantization XVIII, Bulgar. Acad. Sci., Sofia, 2017, 183–190 | DOI | MR

[34] Kuzmina I., Mikeš J., “On pseudoconformal models of fibrations determined by the algebra of antiquaternions and projectivization of them”, Ann. Math. Inform., 42 (2013), 57–64 | MR | Zbl

[35] Levi-Civita T., “Sulle trasformazioni dello equazioni dinamiche”, Ann. Mat. Milano., 24 (1896), 255–300 | Zbl

[36] Lagrange J. L., “Sur la construction des cartes géographiques”, Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin, 1779, no. 4, 637–692

[37] Mikeš J., “On existence of nontrivial global geodesic mappings on $n$-dimensional compact surfaces of revolution”, Proc. Conf. “Differential Geometry and Its Applications” (Brno, 1989), World Scientific, 1990, 129–137 | MR | Zbl

[38] Mikeš J., “Global geodesic mappings and their generalizations for compact Riemannian spaces”, Silesian Univ. Math. Publ., 1 (1993), 143–149 | MR | Zbl

[39] Mikeš J., “Geodesic mappings of affine-connected and Riemannian spaces”, J. Math. Sci., 78 (1996), 311–333 | DOI | MR | Zbl

[40] Mikeš J., Vanžurová A., Hinterleitner I., Geodesic Mappings and Some Generalizations, Palacky Univ. Press, Olomouc, 2009 | MR | Zbl

[41] Mikeš J. et al., Differential Geometry of Special Mappings, Palacky Univ. Press, Olomouc, 2015 | MR | Zbl

[42] Mikeš J., Berezovski V., “Geodesic mappings of affine-connected spaces onto Riemannian spaces”, Colloq. Math. Soc. J. Bolyai., 56 (1992), 491–494 | MR | Zbl

[43] Mikeš J., Berezovski V., Stepanova E., Chudá H., “Geodesic mappings and their generalizations”, J. Math. Sci., 217:5 (2016), 607–623 | DOI | MR | Zbl

[44] Mikeš J., Chudá H., “On geodesic mappings with certain initial conditions”, Acta Math. Acad. Paedagog. Nyházi., 26 (2010), 337–341 | MR | Zbl

[45] Mikeš J., Chudá H., Hinterleitner I., “Conformal holomorphically projective mappings of almost Hermitian manifolds with a certain initial condition”, Int. J. Geom. Meth. Mod. Phys., 11:5 (2014), 1450044 | DOI | MR | Zbl

[46] Najdanović M., Zlatanović M., Hinterleitner I., “Conformal and geodesic mappings of generalized equidistant spaces”, Publ. Inst. Math. Beograd., 98 (112) (2015), 71–84 | DOI | MR | Zbl

[47] Rýparová L., Mikeš J., “On global geodesic mappings of quadrics of revoution”, Proc. 16th Conf. APLIMAT 2017, 2017, 1342–1348 | MR

[48] Rýparová L., Mikeš J., “On geodesic bifurcations”, Geometry, Integrability and Quantization XVIII, Bulgar. Acad. Sci., Sofia, 2017, 217–224 | DOI | MR | Zbl

[49] Rýparová L., Mikeš J., “Bifurcation of closed geodesics”, Geometry, Integrability and Quantization XIX, Bulgar. Acad. Sci., Sofia, 2018, 188–192 | DOI | MR | Zbl

[50] Rýparová L., Mikeš J., “On geodesic bifurcations of product spaces”, J. Math. Sci., 239:1 (2019), 86–91 | DOI | MR | Zbl

[51] Stanković M., Mincić S., Velimirović Lj., Zlatanović M., “On equitorsion geodesic mappings of general affine connection spaces”, Rend. Semin. Mat. Univ. Padova., 124 (2010), 77–90 | DOI | MR | Zbl

[52] Stepanov S., Shandra I., Mikeš J., “Harmonic and projective diffeomorphisms”, J. Math. Sci., 207 (2015), 658–668 | DOI | MR | Zbl

[53] Yano K., Bochner S., Curvature and Betti Numbers, Princeton Univ. Press, 1953 | MR | Zbl