Tangent bundle of a differentiable manifold M even if M isn't orientable

In summary, the conversation discussed the problem of constructing a differentiable structure on a tangent bundle of a differentiable manifold, as proven in Chapter 0 of Riemannian Geometry by Do Carmo. The question was raised about the differential of the overlap map and the speaker ultimately figured it out.
  • #1
JasonJo
429
2
This is a problem many of the grad students have probably encountered, it's in Chapter 0 of Riemannian Geometry by Do Carmo.

Do Carmo proved that the tangent bundle of a differentiable manifold is itself a differentiable manifold by constructing a differentiable structure on TM, where M is a differentiable manifold.

So I wanted to take the differentiable structure that Do Carmo gives in the book, assume that two parametrizations overlap and show that the differential at some point in the overlap has positive determinant.

My real question then is, what exactly IS the differential of the overlap map?

Thanks guys.
 
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  • #2
Nevermind, I got it!

YES!
 

FAQ: Tangent bundle of a differentiable manifold M even if M isn't orientable

1. What is the tangent bundle of a differentiable manifold?

The tangent bundle of a differentiable manifold is a mathematical construction that assigns a vector space to each point on the manifold. It represents all possible directions in which a function defined on the manifold can vary.

2. How is the tangent bundle of a differentiable manifold constructed?

The tangent bundle is constructed by taking the union of all the tangent spaces at each point on the manifold. This creates a new manifold that is twice the dimension of the original manifold.

3. What is the purpose of studying the tangent bundle of a differentiable manifold?

The tangent bundle is an important tool in differential geometry and is used to define important concepts such as vector fields, differential forms, and Lie groups. It also allows for the development of important theorems, such as the Hodge decomposition theorem.

4. Can the tangent bundle of a differentiable manifold be non-orientable?

Yes, the tangent bundle can be non-orientable even if the underlying manifold is orientable. This can occur when the manifold has non-trivial topology, such as a Möbius strip or Klein bottle.

5. How is the orientability of the tangent bundle related to the orientability of the manifold?

If the manifold is orientable, then the tangent bundle will also be orientable. However, if the manifold is non-orientable, the orientability of the tangent bundle will depend on the specific structure of the manifold. In general, the tangent bundle will be non-orientable if the manifold has non-trivial topology.

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