Prolongations of isometric actions to vector bundles

In this paper, we define an isometry on a total space of a vector bundle E by using a given isometry on the base manifold M. For this definition, we assume that the total space of the bundle is equipped with a special metric which has been introduced in one of our previous papers. We prove that the set of these derived isometries on E form an imbedded Lie subgroup Ge of the isometry group I E . Using this new subgroup, we construct two different principal bundle structures based one on E and the other on the orbit space E/Ge. Key words: Fiber bundles, isometry group, vector bundles, principal bundles

Keywords:

Fiber bundles, isometry group, vector bundles principal bundles,

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[1] Scot A, Stuck G. The isometry group of a Compact Lorentz manifold. Inventiones Mathematicae 1997, 129: 239-261. doi: 10.1007/s002220050163.
[2] Alexandrino M, Bettiol G. Lie Groups and Geometric Aspects of Isometric Actions. Switzerland: Springer International Publishing, 2015.
[3] Atiyah MF, Singer IM. The Index of Elliptic Operators. I. Annals of Mathematics Second Series 1968, 87: 484-530. doi: 10.2307/1970715.
[4] Atiyah MF, Hitchin NJ, Drinfelo VG, Manin IM. Construction of instantons. Physics Letters A 1978, 65: 185–187. doi: 10.1016/0375-9601(78)90141-X.
[5] Mas-Colel A. Four Lectures on The Differential Approach to General Equilibrium Theory. In: Ambrosetti A, Gori F, Lucchetti R. (editor) Mathematical Economics. Heidelberg, Berlin: Springer, 2006, pp. 19-43.
[6] Fox RH. On topologies for function spaces. Bulletin of the American Mathematical Society 1945, 51 : 429-432. doi: 10.1090/S0002-9904-1945-08370-0.
[7] Gordon CS, Wilson EN. Isometry groups of Riemannian Solvmanifolds. Transactions of the American Mathematical Society 1988, 307 (1): 245-269. doi: 10.1090/S0002-9947-1988-0936815-X.
[8] Huybrechts B, Lehn M. The Geometry of Moduli Spaces of Sheaves. Germany: Cambridge University Press, 2010.
[9] Ihrig E. The Size of Isometry Groups on Metric Spaces. Journal of Mathematical Analysis and Applications 1983, 96: 447-453. doi: 10.1016/0022-247X(83)90053-7.
[10] Kadioglu H, Fisher R. Metric Structures on Fibered Manifolds Through Partitions of Unity. New Trends in Mathematical Sciences 2016, 4: 266-272. doi: 10.20852/ntmsci.2016218388.
[11] Kobayashi S, Nomizu K. Foundations of Differential Geometry. New York and London: Interscience Publishers, 1963.
[12] Konstantis, Panagiotis, Loose, Frank. A classification of isometry groups of homogeneous 3‐manifolds. Mathematische Nachrichten 2016; 289 (13): 1648-1664. doi: 10.1002/mana.201400292.
[13] Lam, Kai S. Fundamental Principles of Classical Mechanics. A Geometrical Perspective. Singapore: World Scientific, 2014.
[14] Lavrenyuk, Yuraslav. Classification of the local isometry groups of rooted tree boundaries. Algebra and Discrete Mathematics 2007; 2: 104-110.
[15] Luke G. and Mishcenko, Vector Bundles and Their Applications. Boston, USA: Springer, 1998.
[16] Mann LN. Gaps in Dimensions of Isometry Groups of Riemannian Manifolds. Journal of Differential Geometry 1976; 11: 293-298. doi: 10.4310/jdg/1214433426.
[17] Myers SB. and Steenrod N.E. The Group of Isometries of a Riemannian Manifold. The Annals of Mathematics 1939; 40: 400-416. doi: 10.2307/1968928.
[18] Rita S , William Z. The Isometry Groups of Manifolds and The Automorphism Groups of Domains. Transactions of the American Mathematical Society 1987; 301: 413-429. doi: 10.2307/2000347.
[19] Shankar K. Isometry groups of homogeneous spaces with positive sectional curvature. Differential Geometry and its Applications 2001; 14: 57-78. doi: 10.1016/S0926-2245(00)00038-3.
[20] Frank WW. Foundations of Differentiable Manifolds and Lie Groups. New York, USA: Springer-Verlag 2000.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

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