Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Local holonomy groups of induced connections


Author: Mu Chou Liu
Journal: Proc. Amer. Math. Soc. 42 (1974), 272-278
MSC: Primary 53C05
DOI: https://doi.org/10.1090/S0002-9939-1974-0331266-9
MathSciNet review: 0331266
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: There are two naturally induced connections on the tangent bundle, the so called Jacobi connection and the Sasaki connection. By using the elementary theory of systems of linear differential equations, we completely determine the local holonomy group of these two induced connections, and find some relation to the local holonomy group of the manifold itself. There is an induced connection on the vector bundle of linear maps of the fibers. We also investigate the properties of the holonomy group of this bundle.


References [Enhancements On Off] (What's this?)

  • [1] P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew Math. 210 (1962), 73-88. MR 25 #4463. MR 0141050 (25:4463)
  • [2] H. Eliasson, Geometry of manifolds of maps, J. Differential Geometry 1 (1967), 169-194. MR 37 #2268. MR 0226681 (37:2268)
  • [3] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, Interscience, New York, 1963. MR 27 #2945. MR 0152974 (27:2945)
  • [4] O. Kowalski, Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine Angew Math. 250 (1971), 124-129. MR 44 #3244. MR 0286028 (44:3244)
  • [5] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds. I, II, Tôhoku Math. J. (2) 10 (1958), 338-354; ibid. (2) 14 (1962), 146-155. MR 22 #3007; 26 #2987. MR 0112152 (22:3007)
  • [6] J. Vilms, Connections on tangent bundles, J. Differential Geometry 1 (1967), 235-243. MR 37 #4742. MR 0229168 (37:4742)
  • [7] -, Totally geodesic maps, J. Differential Geometry 4 (1970), 73-79. MR 41 #7589. MR 0262984 (41:7589)
  • [8] (a) K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles. I. General theory, J. Math. Soc. Japan 18 (1966), 194-210. MR 33 #1814. MR 0193596 (33:1814)
  • 1. (b) -Prolongations of tensor fields and connections to tangent bundles. II. Infinitesimal automorphisms, J. Math. Soc. Japan 18 (1966), 236-246. MR 34 #743. MR 0200857 (34:743)
  • 2. (c) -Prolongations of tensor fields and connections to tangent bundles. III. Holonomy groups, J. Math. Soc. Japan 19 (1967), 486-488. MR 36 #2084. MR 0219001 (36:2084)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C05

Retrieve articles in all journals with MSC: 53C05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0331266-9
Keywords: Pseudo-Riemannian connection, Sasaki connection, holonomy group, parallel vector field, parallel section, affine fiber map, parallel translation, vector bundle, tangent bundle
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society