Geodesic
On the isometry groups of foliated manifolds
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 216 (2022), pp. 124-132.

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In this paper, we study the isometry group $\mathrm{Iso}_{F}(M)$ of a foliated manifold with an $F$-compact-open topology. This topology depends on the foliation $F$ and coincides with the compact-open topology if $F$ is an $n$-dimensional foliation. If the codimension of the foliation is equal to $n$, then the convergence in this topology coincides with the pointwise convergence. Some properties of the group $\mathrm{Iso}_F(M)$ are proved.
Keywords: manifold, isometry of foliations, topological group, $F$-compact-open topology.
Mots-clés : foliation 
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A. S. Sharipov; G. M. Abdishukurova. On the isometry groups of foliated manifolds. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 216 (2022), pp. 124-132. http://geodesic.mathdoc.fr/item/INTO_2022_216_a12/

[1] Aranson S. Kh., “Topologiya vektornykh polei, sloenii s osobennostyami i gomeomorfizmov s invariantnymi sloeniyami na zamknutykh poverkhnostyakh”, Tr. Mat. in-ta im. V. A. Steklova RAN., 193 (1992), 15–21

[2] Baituraev A. M., “O geometrii sloenii korazmernosti 1”, Itogi nauki tekhn. Ser. Sovr. mat. prilozh. Temat. obz., 144 (2018), 109–116 | MR

[3] Grines V. Z., Levchenko Yu. A., Pochinka O. V., “O topologicheskoi klassifikatsii strukturno ustoichivykh $3$-diffeomorfizmov s dvumernymi bazisnymi mnozhestvami”, Mat. zametki., 97:2 (2015), 318–320

[4] Grines V. Z., Pochinka O. V., “Kaskady Morsa—Smeila na $3$-mnogoobraziyakh”, Usp. mat. nauk., 68:1 (409) (2013), 129–188 | MR

[5] Lukatskii A. M., “Konechnoporozhdennost grupp diffeomorfizmov”, Usp. mat. nauk., 33:1 (199) (1978), 219–220 | MR

[6] Lukatskii A. M., “Issledovanie geodezicheskogo potoka na beskonechnomernoi gruppe Li s ispolzovaniem operatora koprisoedinennogo deistviya”, Tr. Mat. in-ta im. V. A. Steklova RAN., 267 (2009), 204–213 | MR

[7] Narmanov A. Ya., “O geometrii vpolne geodezicheskikh rimanovykh sloenii”, Izv. vuzov. Mat., 1999, no. 9, 26–31

[8] Narmanov A. Ya., Kaipnazarova G. Kh., “Metricheskie funktsii na rimanovykh mnogoobraziyakh”, Uzbek. mat. zh., 2010, no. 1, 11–20 | MR

[9] Narmanov A. Ya., Saitova S. S., “O geometrii orbit vektornykh polei Killinga”, Differ. uravn., 50:12 (2014), 1582–1589

[10] Narmanov A. Ya., Skorobogatov D. A., “Izometricheskie otobrazheniya sloenii”, Dokl. AN RUz., 2004, no. 4, 12–16

[11] Narmanov A. Ya., Sharipov A. S., “O gruppe diffeomorfizmov sloenykh mnogoobrazii”, Itogi nauki tekhn. Ser. Sovr. mat. prilozh. Temat. obz., 181 (2020), 74–83 | MR

[12] Pochinka O. V., “Klassifikatsiya diffeomorfizmov Morsa—Smeila na $3$-mnogoobraziyakh”, Dokl. RAN., 2011, no. 6, 34–37

[13] Rokhlin V. A., Fuks D. B., Nachalnyi kurs topologii. Geometricheskie glavy, Nauka, M., 1977 | MR

[14] Antonelli P. L., Burgelea D., Kahn P. J., “The non-finite type of some diffeomorphism groups”, Topology., 11 (1972), 1–49 | DOI | MR

[15] Arnold V., “Sur la géométrie differentielle des groupes de Lie de dimenzion infnite et ses applications à l'hidrodynamique des uides parfaits”, Ann. Inst. Fourier., 16:1 (1966), 319–361 | DOI | MR

[16] Ehresmann C., “Structures feuilletees”, Proc. 5th Can. Math. Congr., 1961, 109–172

[17] Haefliger A., “Variétés feuilletées”, Ann. Scu. Norm. Super. Pisa., 16 (1962), 367–397 | MR

[18] Helgason S., Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, Toronto, 1978 | MR

[19] Lamoureux G., “Variétés feuilletées. Transversales fermées”, C. R. Acad. Sci. Paris., 270 (1970), 1659–1662 | MR

[20] Langevin R., Rosenberg H., “Feuilletages de codimension 1”, Topology., 16 (1977), 107–111 | DOI | MR

[21] Lawson H., “Foliations”, Bull. Am. Math. Soc., 80 (1974), 369–418 | DOI | MR

[22] Myers S., Steenrod N., “The group of isometries of a Riemannian manifold”, Ann. Math. $(2)$., 40:2 (1939), 400–416 | DOI | MR

[23] Molino P., Riemannian Foliations, Birkhäuser, Boston, 1988 | MR

[24] Narmanov A., Sharipov A., “On the group of foliation isometries”, Meth. Funct. Anal. Topology., 15:2 (2009), 195–200 | MR

[25] Narmanov A., Sharipov A., “On the geometry of submersions”, Int. J. Geom., 3:2 (2014), 51–56 | MR

[26] Omori H., “On the group of diffeomorphisms on a compact manifold”, Proc. Symp. Pure Math., 15 (1970), 167–183 | DOI | MR

[27] Omori H., “Groups of diffeomorphisms and their subgroups”, Trans. Am. Math. Soc., 79:1 (1973), 85–122 | DOI | MR

[28] Reeb G., Sur certains propriétés topoloiques des variétés feuilletées, Hermann, Paris, 1952 | MR

[29] Tamura I., Topology of Foliations: An Introduction, Am. Math. Soc., 2006 | MR

[30] Tondeur P., Foliations on Riemannian Manifolds, Springer-Verlag, New York, 1988 | MR