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I wish to learn about holonomy groups of Riemannian manifolds and the Ambrose- Singer theorem. Please advise some references other than the original paper of Ambrose and Singer.

user90041
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    Did you check http://math.stackexchange.com/questions/427471 ? It gives plenty of references. – Moishe Kohan Oct 04 '13 at 19:12
  • @studiosus Thanks for the link! I had not looked at it. This is indeed useful. – user90041 Oct 04 '13 at 19:59
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    References are given in the last paragraph of https://deaneyang.github.io/blog/blog/math/differential-geometry/riemannian-geometry/2020/08/07/Holonomy.html. The post itself is essentially the Ambrose-Singer theorem for an arbitrary vector bundle with a connection. – Deane Jan 09 '24 at 14:27

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References as follows

HLAVATÝ, VÁCLAV. “The Holonomy Group I. The Curvature Tensor.” Journal of Mathematics and Mechanics, vol. 8, no. 2, 1959, pp. 285–307. JSTOR, http://www.jstor.org/stable/24900561 . Accessed 22 Aug. 2022.

The above is the first of six papers on ‘The Holonomy Group’, by V.Hlavaty, that are all available on JSTOR. Individuals may be able to sign up for a free account with JSTOR.

I suggest starting with a first brief look at pg. 12 of the second paper of the six, see (4.1).

HLAVATÝ, V. “The Holonomy Group II. The Lie Group Induced by a Tensor.” Journal of Mathematics and Mechanics, vol. 8, no. 4, 1959, pp. 597–622. JSTOR, http://www.jstor.org/stable/24900677. Accessed 22 Aug. 2022.

HLAVATÝ, VÁCLAV. “The Holonomy Group III. Metrisable Spaces.” Journal of Mathematics and Mechanics, vol. 9, no. 1, 1960, pp. 89–122. JSTOR, http://www.jstor.org/stable/24900513. Accessed 22 Aug. 2022.

HLAVATÝ, VÁCLAV. “The Holonomy Group IV. The General Ln with Symmetric Connection.” Journal of Mathematics and Mechanics, vol. 9, no. 3, 1960, pp. 453–96. JSTOR, http://www.jstor.org/stable/24900484. Accessed 22 Aug. 2022.

HLAVATÝ, VÁCLAV. “The Holonomy Group V. Weyl Space W 4 , First Part.” Journal of Mathematics and Mechanics, vol. 10, no. 2, 1961, pp. 317–48. JSTOR, http://www.jstor.org/stable/24900828. Accessed 22 Aug. 2022.

HLAVATÝ, V. “The Holonomy Group VI. Weyl Space W 4 , Second Part.” Journal of Mathematics and Mechanics, vol. 11, no. 1, 1962, pp. 35–59. JSTOR, http://www.jstor.org/stable/24900845. Accessed 22 Aug. 2022.

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