What are the minimum pre-requisites in analysis (differential geometry) required to study Lie-groups? And for that material, what are some good references?

I have done basic courses in Metric spaces, Topology, Complex analysis etc. and Linear Algebra and Functional analysis. I also have some knowledge of Curves and surfaces ( $\mathbb{R^3}$ ).

I have to do a reading in Lie groups and, perhaps, later continue with its representation theory. Though it is (probably?) an algebraic study, I would still like to know the role played by Lie groups and algebras in Geometry too.

Ideally I would prefer to have a brief but sufficiently rigorous introduction in Differential geometry so that I may continue with the study of Lie groups without hindrance. For that if there is a reference recommendation then I would be really thankful.If there exist some lecture notes serving this purpose,then that would be great too.)

Thanks in advance!

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    $\begingroup$ i think the following notes of Wolfgang Ziller on Lie groups are exactly what you are looking for! math.upenn.edu/~wziller/math650/LieGroupsReps.pdf $\endgroup$ – Christos Jan 31 '16 at 13:14
  • $\begingroup$ @Christos It already assumes the stuff I am seeking. There is the relevant bibliography, though it is scattered into a few books . I would prefer somewhere with all the relevant things; e.g some book where I can read all the differential geometry I will need to study Lie groups. $\endgroup$ – Chiha Bakz Jan 31 '16 at 13:24
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    $\begingroup$ Brian C Hall's Lie Groups/Algebras text gives an introduction to the subject that makes heavy use of linear algebra knowledge. However, if you really want a differential topology/geometry approach to manifolds first, I'd recommend Loring Tu's text on Manifolds as a first stop. $\endgroup$ – Ben Grossmann Jan 31 '16 at 13:40
  • $\begingroup$ @Omnomnomnom: I had a look at that book (Loring Tu). It seems quite nice and readable. Though it has a lot of material (around 100 pages ) before it comes to Lie groups part. Please migrate your comment to the answers. If I don't stumble across any answer mentioning a more concise text, then this seems like the best candidate for my need. $\endgroup$ – Chiha Bakz Feb 1 '16 at 17:18
  • $\begingroup$ @ChihaBakz done. I hope you find what you're looking for. $\endgroup$ – Ben Grossmann Feb 1 '16 at 21:03

My answer, as in the comments above, with some elaboration:

I recommend Loring Tu's "An Introduction to Manifolds" as an accessible first stop. It does have a lot of reading material before the Lie Groups part, but you can skip through a lot of the material leading into that.

To get through the minimal background required for what you want, I recommend the following course of action: skim through chapters 1-4. Read enough to understand what the questions are actually asking, and perhaps solve some of the easier ones. A lot of this is referred to, but you can get by just understanding the big ideas. You probably have some understanding of this material coming in, anyway.

Read chapters 5, 6, 8, 12, and 14. You are now set for the Lie-groups/algebras sections, which is to say 15-16.

The other material is worth a read if you're into the more topological/algebraic side of things (especially if you're working your way towards homology/cohomology and the like). However, I'm pretty sure none of it is necessary for an understanding of Lie theory basics.


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