I am sorry if my question is stupid (or very hard) or common knowledge, or should be placed at math.stackexchage.com.
As long as a math student read the definition of Lie group, several natural questions appear instantly:


Is it true, that any compact manifold admits Lie group structure? (NO)


Is it true, that there exists compact manifold that admits different Lie group structures? (YES)


Answers to these questions can be found here - [Lie Groups and Manifolds](https://mathoverflow.net/questions/5262/lie-groups-and-manifolds)


But I was not able to find the answer for third most natural question:
Which connected compact manifolds admits unique Lie group structure?
(As pointed out by YCor, without the connectedness assumption, there are trivial non-uniqueness examples.)


Thanks a lot for your answers!



Connected compact manifolds with unique Lie group structure
Your question seems to be of a tall order!


There are many,vmany necessary conditions involving the entire
spectrum of algebraic topology such as homotopy and cohomology.
[This 3rd edition book](https://www.degruyter.com/view/product/185231) by Kakl H. Hofmann and Sidney A. Morris *on compact groups* is full of these. For instance if you asked: *which compact connected n-dimensional manifold can support an
abelian Lie group,* one would tell you "only $\mathbb{T}^n =(\mathbb{R}/\mathbb{Z})^n$" but without the qualification "unique". There are tons of
discontinuous automorphisms of $\mathbb{T}$, and so you have millions of group topologies (all isomorphic!). 


By contrast,
the $3$-dimensional real projective space supports only one compact
Lie group topology; namely, that of $SO(3)$. This has to do with the
automatic continuity of algebraic automorphisms of certain compact
connected Lie groups (see loc.cit., Corollary 5.66 and Exercise E5.21
following it, page 168). For more stuff see (just as a for instance)
8.59, 8.83, 9.59, A3.90, A3.92.


Life is not so simple that we can hope for a positive answer to your
question as it stands. However, information in the area is around so that
as soon as you pose a question of a *more specific* nature on
the topic of compact connected manifolds and Lie groups, there
is hope for an answer.


