Tangent Bundle of Product is diffeomorphic to Product of Tangent Bundles

In summary, the conversation discusses the struggle to show that T(MxN) is diffeomorphic to TM x TN, with the suggestion to use the product of projection maps from MxN to M and N to form the diffeomorphism. The conversation also mentions the importance of understanding previous material and the need for a precise application of all definitions.
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Amateur659
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My apologies if this question is trivial. I have searched the forum and haven't found an existing answer to this question.

I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM x TN.

My intuition is that I could set up the product of projection maps from MxN to M and MxN to N to form the diffeomorphism. However, I seem to be stuck on the implementation (which probably indicates gaps in my understanding of previous material).

Thanks for your assistance.
 
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  • #2
This is the kind of question that requires (only) a precise application of all definitions. The sets are clearly identical:
$$
T(M\times N)= \bigcup_{(p,q)\in M\times N}\{(p,q)\} \times T_{(p,q)}(M\times N) = \bigcup_{p\in M} \{p\}\times T_{p}(M)\times \bigcup_{q\in N} \{q\}\times T_{q}(N)=T(M) \times T(N)
$$
Next, we need a differentiable structure on both. Differentiability is a local property. So we can pick a point ##(p,q)\in U\times V## from a contractible neighborhood. By defining it as ##U\times V## we already fixed the topology, namely the product topology. Now gather all local diffeomorphisms ##TU=U\times \mathbb{R}^n\, , \,TV=V\times \mathbb{R}^m## and combine them to a diffeomorphism ##T(U\times V)= U\times V \times \mathbb{R}^{n+m}.##
 
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1. What is the Tangent Bundle of Product?

The Tangent Bundle of Product refers to the collection of all tangent spaces of a product manifold. It is a vector bundle that is formed by taking the direct product of the tangent spaces of each factor in the product manifold.

2. What does it mean for the Tangent Bundle of Product to be diffeomorphic to the Product of Tangent Bundles?

When the Tangent Bundle of Product is diffeomorphic to the Product of Tangent Bundles, it means that there exists a smooth bijective map between the two bundles that preserves the differential structure. In simpler terms, the two bundles are essentially the same in terms of their differentiability properties.

3. Why is the Tangent Bundle of Product diffeomorphic to the Product of Tangent Bundles?

The Tangent Bundle of Product is diffeomorphic to the Product of Tangent Bundles because both bundles are constructed using the same underlying product manifold. The direct product of tangent spaces of each factor in the product manifold results in a bundle that is isomorphic to the individual tangent bundles of each factor.

4. What is the significance of the Tangent Bundle of Product being diffeomorphic to the Product of Tangent Bundles?

This result is significant because it allows for the simplification of certain calculations and proofs in differential geometry. It also provides a deeper understanding of the relationship between differentiable structures on manifolds.

5. Can the Tangent Bundle of Product be diffeomorphic to the Product of Tangent Bundles for non-product manifolds?

No, the Tangent Bundle of Product can only be diffeomorphic to the Product of Tangent Bundles for product manifolds. This is because the construction of the two bundles relies on the direct product structure of the underlying manifold.

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