A Lie group is a smooth manifold whose tangent space at its origin is its Lie algebra. Taking an example for lie group such as SL(2), and due to above facts we should then be able to translate the abstract study of manifolds to some concrete computations dealing only with numbers. What are some references that exactly do that?

Just to give some examples, we know beforehand that tangent space will contain matrices with so and so property for some particular Lie group. And then we compute to show exactly that. Similarly, can we concretely define geodesics on Lie groups totally in a computational manner?

References are fine. In case, there are certain abstract manifold concepts that cannot be computationally translated in terms of Lie group, please mention them and say some lines as to why. A philosophical answer is also sufficient.

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    $\begingroup$ This should be very easy to find. If you understand the abstract stuff it should be easy to compute the associated Lie algebra( as a subset of matrices) relatively easily. Computing matrix exponentials is also done in loads of places. I think Stillwell's "Naive Lie Theory" does a good amount of this at a pretty basic level for lots of examples including the computation you gave. Computing matrix exponentials is done in loads of books. Some of this is in Artin as well. $\endgroup$ – PVAL-inactive Jul 1 '16 at 1:55
  • $\begingroup$ Generally in topology, the only time you leave the realm of being able to write your maps in explicit formulas is when you envoke Baire category theory stuff (e.g. Sard's theorem and transversality type results) or contraction mapping principle (e.g. existence of flows for short time given a vector field.). $\endgroup$ – PVAL-inactive Jul 1 '16 at 1:58
  • $\begingroup$ @PVAL Are there other good books that explicitly compute charts and geodesics for the sake of it? I understand that in pure maths, abstraction is considered correct way of thinking and computation is frowned upon. But there is a section of mathematics that believes in developing theory by proving computational results. Unfortunately i have not been able to lay my hands on books following that direction. $\endgroup$ – user_1_1_1 Jul 1 '16 at 14:00

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