مروری بر عمل های با نقص همگنی یک

نوع مقاله : مقاله علمی - مروری

نویسندگان

1 دانشگاه زنجان، دانشکده علوم پایه، گروه ریاضی

2 دانشگاه بین المللی امام خمینی، گروه ریاضی محض

چکیده

در این مقاله، پس ارائه تاریخچه ای از عمل های با نقص همگنی یک، نتایج پژوهش های انجام شده در زمینه رده بندی عمل های با نقص همگنی یک بر خمینه های ریمانی و شبه ریمانی با تقریب هم ارزی مداری آورده شده است. همچنین مسئله های باز پژوهشی موجود در این زمینه معرفی شده اند.

کلیدواژه‌ها

موضوعات


[1] Adams, S., The isometry group of a compact Lorentz manifold (I), Inventions Mathematicae,
129 (1997), 239–261.
[2] Ahmadi, P., Cohomogeneity one dynamics on three dimensional Minkowski space, Submited.
[3] Ahmadi, P., Cohomogeneity one three dimensional anti-de Sittere space, proper and nonproper
actions, Differential Geometry and its Applications, 39 (2015), 93–112.
[4] Ahmadi, P. and Kashani, S. M. B., Cohomogeneity one Minkowski space R
n
1 , Publicationes
Mathematicae Debrecen, 78 (2011), no. 1, 49–59.
[5] Ahmadi, P., Kashani, S. M. B., Abedi, H., Cohomogeneity one de Sitter Space S
n
1 , Acta Mathematica Sinica, 26 (2010), no. 10, 1915–1926.
[6] Alekseevsky, A. V., Alekseevsky, D.V., G-manifolds with one dimensional orbit space, Advances in Soviet Mathematics, 8 (1992), 1–31.
[7] Alekseevsky, D. V., Compact quaternion spaces, Functional Analysis and Its Applications, 2
(1968), 106–114.
[8] Alekseevsky, D.V. On a proper action of a Lie group, Uspekhi Matematicheskikh Nauk, 34
(1979), 219–220.
[9] Alekseevsky, D. V., Riemannian manifolds of cohomogeneity one, Colloquia Mathematica
Societatis János Bolyai, 56 (1989), 9–22.
[10] Atiyah, M., Berndt, J., Projective planes, Severi varieties and spheres: Surveys in Differential
Geometry (VIII), Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck, Somerville, MA,
1–27, 2003.
[11] Bérard Bergery, L., Sur de nouvelles variétés riemanniennes d’Einstein, Institut Élie Cartan,
4 (1982), 1–60.
[12] Berndt, J. Lie group actions on manifolds, Lecture note of a graduate course given at Sofia
University, Tokyo, 2002.
[13] Berndt, J. and Brück, M. Cohomogeneity one actions on hyperbolic spaces, Journal für die
Reine und Angewandte Mathematik, 541 (2001), 209–235.
[14] Berndt, J., Diaz-Rawos, J. C., Vanae, M. J., Cohomogeneity one actions on Minkowski spaces,
arXive:1410.1700 [math.DG].
[15] Berndt, J. and Tamaru, H., Homogeneous codimension one foliations on noncompact symmetric spaces, Journal of Differential Geometry, 63 (2003), 1–40.
[16] Berndt, J. and Tamaru, H. Cohomogeneity one actions on noncompact symmetric spaces with
a totally geodesic singular orbit, Tohoku Mathematical Journal, 2 (2004), no. 56, 163–177.
[17] Berndt, J., Tamaru, H., Cohomogeneity one actions on noncompact symmetric spaces of rank
one, Transactions of the American Mathematical Society, 359 (2007), no. 7, 3425–3438.
[18] Berndt, J., Tamaru, H., Cohomogeneity one actions on noncompact symmetric spaces of noncompact type, Journal für die Reine und Angewandte Mathematik, 683 (2013), 129–159.
[19] Berndt, J., Dom’inguez-V’azquez, M., Cohomogeneity one actions on some noncompact symmetric spaces of rank two (English summary), Transformation Groups, 20 (2015), no. 4, 921–
938.
[20] Borel, H., Some Remarks about Lie groups transitive on sphere and tori, Bulletin of the American Mathematical Society, 55 (1949), 580–587.
[21] Cartan, E., Familles de surfaces isoparamétriques dans les espaces à courbure constante, Annali di Matematica Pura ed Applicata, IV. Ser. 17 (1938), 177–191.
[22] D’Atri, J. E., Certain isoparametric families of hypersurfaces in symmetric spaces, Journal of
Differential Geometry, 14 (1979), 21–40.
[23] Gromov, M., Rigid Transformation Groups, In: D. Bernard, Y. Choquet-Bruhat (eds.), Geometrie Differentielle, Herrmann, Paris, 1988.
[24] Hsiang, W. Y. and jun Lawson, H.B. Minimal submanifolds of low cohomogeneity, Journal
of Differential Geometry, 5 (1971), 1–38.
[25] Iwata, K. Classiffication of compact transformation groups on cohomology quaternion projective spaces with codimension one orbits, Osaka Journal of Mathematics, 15 (1978), 475–508.
[26] Iwata, K. Compact transformation groups on rational cohomology Cayley projective planes,
Tohoku Mathematical Journal, II. Ser. 33 (1981), 429–442.
[27] Kobayashi, S. Transformation Groups in Differential Geometry, Volume 70 of Ergebnisse der
Mathematik und iher Grenzgebiete, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
[28] Kollross, A., A classification of hyperpolar and cohomogeneity one actions, Transactions of
the American Mathematical Society, 354 (2002), 571–612.
[29] Koszul, J. L., Sur la forme hermitienne des espaces homogènes complexes, Canadian Journal
of Mathematics, 7 (1955) , 562–576.
[30] Kowalsky, N., Actions of non-compact simple groups on Lorentz manifolds and other geometric manifolds, Thesis: University of Chicago, 1994.
[31] Lee, J. M., An Introduction to Smooth Manifolds, Springer-Verlag, 3rd edn., 2013.
[32] Levi Civita, T., Famiglie di superficie isoparametriche nell’ordinario spazio euclideo, Atti della
Accademia Nazionale dei Lincei, 6 (1937), no. 26, 355–362.
[33] Montgomery, D., Samelson, H., Transformation groups on spheres, Annals of Mathematics,
44 (1943), 457–470.
[34] Montgomery, D., Samelson, H., Yang, T., Groups on E
n with (n − 2)-dimensional orbits,
Proceedings of the American Mathematical Society, 7 (1956), 719–728.
[35] Montgomery, D., Zippin, L., A class of transformation groups in E
n
, American Journal of
Mathematics, 63 (1941), 1–8.
[36] Mostert, P. S., On a compact Lie group acting on a manifold, Annals of Mathematics, 65
(1957), 447–455.
[37] O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New
York, (1983).
[38] Palais, R. S., On the existence of slices for actions of non-compact Lie groups, Annals of
Mathematics, 73 (1961), 295–323.
[39] Podesta, F., Spiro, A., Some topological properties of cohomogeneity one Riemannian manifolds of negative curvature. Annals of Global Analysis and Geometry, 14 (1996), 69–79.
[40] Segre, B., Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque
numero di dimensioni, Atti della Accademia Nazionale dei Lincei, 6 (1938), no. 27 , 203–207.
[41] Straume, E., Compact connected Lie transformation groups on sphere with low cohomogeneity
(I), Memoirs of the AMS, 119 (1996), no. 569.
[42] Takagi, R. On homogeneous real hypersurfaces in a complex projective space, Osaka Journal
of Mathematics, 10 (1973), 495–506 .
[43] Uchida, F., Classiffication of compact transformation groups on cohomology complex projective spaces with codimension one orbits, Japanese Journal of Mathematics, 3 (1977), 141–189.
[44] Wang, Q. M., Isoparametric hypersurfaces in complex projective spaces, Differential Geometry and Differential Equations, Proceedings of 1980 Beijing symposium, 3 (1982), 1509–1523.
[45] Zimmer, R. J., On the automorphism group of a compact Lorentz manifold and other geometric
manifolds, Inventions Mathematicae, 89 (1986), 411–424.