Reduced bundles and global sections of associated bundle

I am following Kobayashi and Nomizu's book Foundations of differential geometry volume 1 (page no $$57$$) Proposition 5.6.

The structure group of a principal bundle $$P(M,G)$$ is reducible to a closed subgroup $$H$$ of $$G$$ if and only if the associated bundle $$E=(P\times G/H)/G\rightarrow M$$ has a global section. (Here we use the notation $$Q(M,H)$$ for the reduced bundle.)

One direction I was able to understand. Given that $$G$$ is reducible to $$H$$, I was able to produce a global section for $$E\rightarrow M$$.

For the other direction, I was able to understand everything except that I could not prove $$Q$$ is an immersed submanifold.

If the following result is true, then I am done.

Is the inverse image of an immersed submanifold an immersed submanifold under smooth submersion?

I know the above result is true for embedded submanifold (Using transversality) but I am not sure about the result for immersed submanifold.

1 Answer

Yes - use that an immersion is locally an embedding to reduce to the case of an embedded submanifold.

Added:

Let $$U$$ be the immersed submanifold with preimage $$V$$. Cover $$U$$ by embedded submanifolds $$U_i$$. The preimages $$V_i$$ are embedded and cover $$V$$, so $$V$$ is an immersion.

• Thanks @Ben for the answer!..I think you are talking about local embedding theorem. But to use the result to boil it down to the embedded case I am needing another result that I am not sure of.. I need union of embedded submanifolds is an embedded submanifold... Is that true? Or we have to approach in a different way? I am Sorry if my remark is stupid. Jan 30, 2019 at 14:36
• @ADITTYACHAUDHURI The union of embedded manifolds is only an immersed manifold. But that good enough right?
– Ben
Jan 30, 2019 at 15:51
• @ADITTYACHAUDHURI By the way I added another line of explanation.
– Ben
Jan 30, 2019 at 17:53
• @ Ben Thanks! Actually I never came across about the result that the union of embedded submanifiolds is an immersed submanifold. Since there are examples where union of submanifolds is not a submanifold I never thought in that direction.Can you please refer me a literature (or give a brief outline ) for the proof of the fact that union of embedded submanifolds is an immersed submanifold? I could not see directly that local embedding theorem is equivalent to "This Result". Sorry in advance if my remark is stupid! Jan 31, 2019 at 1:07
• @ADITTYACHAUDHURI Here by "union of embedded submanifolds" I just mean it is a union of open subsets which are embedded. That is, given $M \to N$ and $U_i$ covering $M$ such that $U_i \to N$ are embeddings, I am saying "$M$ is a union of embedded manifolds". But this is just the condition of locally being an embedding.
– Ben
Jan 31, 2019 at 5:44