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I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are analytic, for example the function given here. Moreover from this particular question asked on MSE we also know that the set of all Non-analytic smooth functions are dense in the space of continuous function on compact interval in $sup$-norm. So my questions are

Questions :

1) Suppose we have a smooth manifold. Does it always has an analytic structure ?

2) Like approximation of continuous functions by polynomials, does it make sense to say something similar about ${approximation}$ $by$ or $on$ manifolds ?

3) Suppose the answer to 2nd question is affirmative. Then since the set of all non-analytic smooth functions are dense as described above, can we say that any smooth manifold can be $approximated$ by manifolds having an analytic structure ?

I have thought about the 1st problem and I think that the answer is NO. But I wasn't able to produce a counter-example. I was not able to proceed by taking a non-analytic smooth function and to do something with that. As for 2) and 3) I don't have any idea about them.

Thanks in advance.

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  • was asked on MO: http://mathoverflow.net/questions/8789/can-every-manifold-be-given-an-analytic-structure. Apparently the answer is yes.
    • – Qiaochu Yuan Feb 02 '15 at 07:16
  • @QiaochuYuan : Thanks a lot for that link. that is exactly what i was looking for, for 1). please let me know about any ideas you have for 2) and 2) also. – wanderer Feb 02 '15 at 07:35
  • You should take a look at Chapter 2 of Hirsch's "Differential Topology", in particular section 5. It provides references for some things you're looking for. –  Feb 02 '15 at 07:44