Differential Geometry: Finding Integral Manifolds

In summary, the conversation is about someone seeking help with finding integral manifolds in differential geometry. The other person suggests reading up to page 503, where a method for finding integral manifolds is explained with an example. The concept of integral manifolds is also briefly defined as all manifolds where there is a linear map between the tangent space and the given distribution at each point.
  • #1
Sephi
6
0
Hi people,
I'm learning differential geometry in a book (Intro to smooth manifolds, by John Lee) and I have some difficulties with the tangent distributions.
Actually, I don't know what to do if, given a distribution spanned by some vectors fields, I want to find its integral manifolds.
Can someone help me ?
 
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  • #2
I either do not understand your question or I may be stating something that you already know...
Integral manifolds of a given distribution are all manifolds [tex]M[/tex] for which [tex]\forall p \in M[/tex] there is a linear map between the tangent space and the distribution at that point.
 
  • #3
Have you read up to page 503? There, it is remarked that embedded in the proof of Frobenius' theorem is a technique for finding integral manifolds and an example illustrating the method is given.
 

1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves, surfaces, and higher-dimensional manifolds using tools from calculus and linear algebra. It provides a framework for analyzing and understanding the geometry of spaces that may be curved or have more than three dimensions.

2. What are integral manifolds?

Integral manifolds are submanifolds that are invariant under the flow of a given differential equation. They can be thought of as the trajectories of a dynamical system, and they play a crucial role in understanding the behavior of the system as a whole.

3. Why is finding integral manifolds important?

Finding integral manifolds allows us to understand the long-term behavior of a dynamical system and make predictions about its future behavior. This is especially important in fields such as physics, engineering, and economics, where understanding and predicting the behavior of systems is crucial.

4. What techniques are used to find integral manifolds?

There are several techniques used to find integral manifolds, including the method of characteristics, the method of Frobenius, and the method of Lie series. These techniques involve using differential equations and linear algebra to find solutions that satisfy the conditions for being an integral manifold.

5. How are integral manifolds related to differential equations?

Integral manifolds are intimately related to differential equations, as they are the solutions that remain invariant under the flow of the equations. In fact, finding integral manifolds can often be reduced to solving a system of differential equations. Conversely, understanding the behavior of integral manifolds can provide insights into the behavior of the underlying differential equations.

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