Differential Forms on Smooth Manifolds

In summary, the conversation discusses the existence of vector fields on a smooth manifold M that satisfy certain conditions with locally chosen 1-forms. The possibility of extending these vector fields to the entire manifold is also considered. The conversation was inspired by the proof of Frobenius integrability theorem in Spivak's Volume 1, where locally defined vector fields are used. The existence of globally defined vector fields is questioned, as they may not exist on certain manifolds such as the sphere.
  • #1
daishin
27
0
Let M be a smooth manifold. Locally we can choose 1-forms [tex]\omega[/tex][tex]^{1}[/tex],[tex]\omega[/tex][tex]^{2}[/tex],...[tex]\omega[/tex][tex]^{n}[/tex] whish span M[tex]^{*}_{q}[/tex] for each q. Then are there vector fields X[tex]_{1}[/tex], X[tex]_{2}[/tex], ...,X[tex]_{n}[/tex] with [tex]\omega[/tex][tex]^{i}[/tex](X[tex]_{j}[/tex])=[tex]\delta^{i}_{j}[/tex]? Here [tex]\delta^{i}_{j}[/tex] is Kronecker delta.
By vector fields, I meant vector fields on M.
I think there are such vector fields on small neighborhood B in M.(since M* is locally
trivial, we can think of M* restricted to B as B X R^n. And we can find such 1-forms w_1, w_2,...w_n which span M* at each p in B. And of course we can find vector fields X[tex]_{1}[/tex], X[tex]_{2}[/tex], ...,X[tex]_{n}[/tex] on B such that
[tex]\omega[/tex][tex]^{i}[/tex](X[tex]_{j}[/tex])=[tex]\delta^{i}_{j}[/tex].
But I am wondering if we can extend this vector fields to whole of M.
 
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  • #2
You started with 1-forms which were chosen locally. So before answering your latter question, you should think about whether you can choose the 1-forms globally on M.
 
  • #3
I think we can always find globally defined 1-forms w_1, w_2,...w_n on M which in some small neighborhood B, they span M* for each p in B. If not please correct me.
My question came from the proof of Frobenius integrability theorem in Spivak Volume 1.
It is a chapter 7 Theorem 14. He starts the proof with locally defined 1-forms w_1,w_2,...,w_n. But in the proof he says:Let X_1, X_2,... X_n be the vecor fields with
w_i(X_j)= delta^i_j. Here, I think he is referring vector fields on M.
 
  • #4
In general such X does not exist since e.g. on sphere you cannot construct nowhere vanishing vector field. So I think the proof refers to locally defined fields.
 

1. What are differential forms on smooth manifolds?

Differential forms on smooth manifolds are mathematical objects used in differential geometry to describe and study the behavior of smooth functions on curved spaces, known as smooth manifolds. They are used to generalize concepts such as integration, differentiation, and Stokes' theorem to these curved spaces.

2. How are differential forms different from traditional calculus objects?

Differential forms are different from traditional calculus objects, such as functions and vectors, because they are defined in terms of the tangent space of the manifold, rather than the manifold itself. This allows them to capture the local behavior of a function on the manifold, rather than just its global behavior.

3. What is the significance of the exterior derivative in differential forms?

The exterior derivative is a fundamental operation in differential forms that allows us to differentiate forms of different degrees. It is an extension of the concept of differentiation from single-variable calculus, and is used to define the important notion of closed and exact forms.

4. How are differential forms used in physics?

Differential forms are used extensively in physics, particularly in the study of classical and quantum field theory. They provide a powerful framework for describing physical phenomena on curved spaces, such as spacetime in general relativity, and are also used in other areas of physics, such as electromagnetism, fluid dynamics, and thermodynamics.

5. What are some applications of differential forms in mathematics?

Differential forms have numerous applications in mathematics, including in geometry, topology, and analysis. They are used to study and classify manifolds, to define and solve differential equations, and to prove important theorems in these fields, such as the Poincaré lemma and de Rham's theorem.

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