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Motivated by this MO question we ask the following two questions:

1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each fibre(tangent space) gives a simple Lie algebra?

2)What is an example of a compact manifold $M$ which admit at least one smooth (1,2) tensor $\omega$ which restriction to each fibre(tangent space) gives a simple Lie algebra?

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    $\begingroup$ My understanding is that there are, up to isomorphism, precisely two 2-dim Lie algebras: the abelian one and one with $[x,y] = x$. But neither is simple. So it seems as though any surface gives an example of 1). That said, you mention in the linked MO question that every parallelizable manifold has the "tensorial property", so perhaps I'm missing something... $\endgroup$
    – Jason DeVito - on hiatus
    Aug 20, 2015 at 4:52
  • $\begingroup$ @JasonDeVito Thanks for your comment. According to your comment I realize that I should add "n>2" in this and MO question. is this what you mean? $\endgroup$
    – Ali Taghavi
    Aug 20, 2015 at 8:57
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    $\begingroup$ Well, perhaps that is what I mean :-). On a related note, I don't think there are any simple Lie algebras in dimension 4,5,6, 7, 9, 11, ..., so there should be plenty of examples of 1) in these dimensions. On the other hand, every orientable three manifold is parallelizable, so these are all examples of 2). $\endgroup$
    – Jason DeVito - on hiatus
    Aug 21, 2015 at 13:56
  • $\begingroup$ @JasonDeVito Thanks for your information on dimension 4,5,6.. $\endgroup$
    – Ali Taghavi
    Aug 22, 2015 at 20:54

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