1)Is there an example of a non trivial principal bundle $P(M,G)$ such that the total space admit a foliation which all leaves are diffeomorphic to $M$?
2) Is there an example of a non trivial principal bundle $P(M,G)$ with the property that every integrable distribution on $M$ can be lifted to an integrable distribution on the total space?
G is a Lie group and acts smoothly on M i think the integrability structure on M can be lifted on the total space E where M=E/G
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fibre bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fibre bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action.
https://books.google.com.tr/books?id=A7crD-orhV8C&pg=PA588&lpg=PA588&dq=in+the+principal+bundle+integrability+structure+can+be+lifted+on+total+space&source=bl&ots=Ry1a-Hfm-X&sig=NrMdE-1lhx0ckJM0eOcJJgl-9vk&hl=tr&sa=X&ved=0CDMQ6AEwA2oVChMIgJDYwvqcyQIVQ1osCh3f_QXJ#v=onepage&q=in%20the%20principal%20bundle%20integrability%20structure%20can%20be%20lifted%20on%20total%20space&f=false
https://books.google.com.tr/books?id=sIsJcxfzgeYC&pg=PA271&lpg=PA271&dq=in+the+principal+bundle+integrability+structure+can+be+lifted+on+total+space&source=bl&ots=Q4eME6C7tK&sig=ojKuRA8RdDnYvMsAxNLy7eqmmio&hl=tr&sa=X&ved=0CGIQ6AEwCWoVChMIgJDYwvqcyQIVQ1osCh3f_QXJ#v=onepage&q=in%20the%20principal%20bundle%20integrability%20structure%20can%20be%20lifted%20on%20total%20space&f=false
I can't answer Question 1, but I bet there's an example. A rather trivial positive answer to Question 2 is given by the principal Z_2 bundle with total space and base space equal to the unit circle S1 in the complex plane, and projection z ----> z2 . But I don't know an example with connected G.
These are interesting questions. Try asking in www.mathoverflow.net
Q1: I do not quite understand the question. Let $\pi : P \to M$ be the principal fibration. If there is a submanifold of $P$ which projects diffeomorphically onto $M$ by $\pi$, this submanifold is a section of $\pi$, thus the fiber bundle is trivial.
Q2: What do you mean by "lifted"? If you allow the dimension to be changed, this is always true: If $S\subset M$ is any submanifold, take $\pi^{-1}(S)\subset P$.
If you want to keep the dimension, just take $S = M$; a lift of $S$ in this sense would be a section of $\pi : P \to M$, hence $P$ is a trivial bundle.
Best regards
Jost
Dear Prof. Hirsch Thank you very much for your answer. I try to follow the question both in RG and MO.
Dear Prof. Eschenburg Thank you very much for your answer. In my first question I do not assume that the leaves are projected diffeomorphicaly to the base space. I search for an example of nontrivial bundle such that the totall space is foliated by leaves diffeomorphic to M, but not in a canoncal manner. May be the following could be a possible candidate: If we could have a non parallelizable manifold M which TM is diffeomorphic to MxR^n, then may be the frame bundle L(M) could be a possible candidate.
Regarding my second question: Assume that we have a distribution D on M, we say that a distribution D' on P is a lifting of D if they have the same dimension and D' is \pi- related to D.
In your last statement, you consider the full dimensional distribution on M. So a lifting D' is nothing but a connection on P. Now I do not understand that why integrability of D' imply that all leaves are diffeomorphic to M (via \pi)? May be I am missing some thing?
Dear Cenap Thank you very much for the links. But I still do not understand why the bundle is trivial?
Best regards
Ali Taghavi
1) Let's take a point p of M, and the fiber F above it. Choose a leave L whose intersection with F is the point q. As it is diffeomorphic to M, to a small enough neighboorhood U of p must correspond a neighboorhood V of q in L. If the intersection of V with F is a point, then L is a section, which is impossible. Therefore, L must contain F. Making the same reasoning with the same leave and different fibers, we see that L is a union of fibers. If such an example exists, M itself must be a fiber bundle with the same typical fiber, and each of its fibers is diffeomorphic to a fiber of P. I can't go further, but that already excludes many non trivial fiber bundles like the Hopf ones.
Another possiblity is that the intersection of L with F is a finite set of discrete points. In this case, the bundle of orthonormal frames on S^2, is pehaps such an example that is analogous to the belt trick.
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I received some answers to the first question in MO as follows
http://mathoverflow.net/questions/224841/some-examples-of-non-trivial-principal-bundles
Dear Ali:
"In your last statement, you consider the full dimensional distribution on M. So a lifting D' is nothing but a connection on P. Now I do not understand that why integrability of D' imply that all leaves are diffeomorphic to M (via \pi)? May be I am missing some thing?"
No, you are right; I missunderstood the notion of lifting. What you wrote contains the full answer to the case of full dimension: You get a connection on P which is integrable - this is the same as "flat". Precisely flat connections have the requred property. In a flat connection the local holonomy is trivial; all holonomy comes from the fundamental group of M.
The easiest example was already mentioned in the discussion: the double cover S^1 \to S^1 (by squaring) with fibre \pm1 = Z/2Z.
Best regards
Jost
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