A problem about integral curves on a manifold

In summary, the conversation discusses how to demonstrate that if c(t) is an integral curve of a smooth vector field X on a smooth manifold M with c'(t_0)=0 for some t_0, then c is a constant curve. One way to prove this is by using the flow of X and showing that X is invariant under its own flow. Another possible approach, based on the fundamental theorem on EDO, was initially thought to not work but has since been solved.
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quasar987
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I must demonstrate in two ways that if c(t) is an integral curve of a smooth vector field X on a smooth manifold M with c'(t_0)=0 for some t_0, then c is a constant curve.

I found one way: If [itex]\theta[/itex] denotes the flow of X, then because X is invariant under its own flow, we have

[tex]c'(t)=X_{c(t)} = (\theta_{t-t_0})_*X_{c(t_0)}=(\theta_{t-t_0})_*c'(t_0)=0[/tex]

for all t.

Does anyone see another way?
 
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  • #2
(Solved. I had initially thought that and argument based directly on the fundamental theorem on EDO does not work, but it does.)
 

Related to A problem about integral curves on a manifold

What is a manifold?

A manifold is a mathematical space that is locally similar to Euclidean space. It is a type of geometric structure that can be described by a set of coordinates and functions that define how points in the space are related to one another.

What are integral curves on a manifold?

Integral curves on a manifold are a way of representing the paths that a vector field takes on the manifold. They are curves that are tangent to the vector field at every point, and they show the direction in which the vector field is changing at each point along the curve.

Why are integral curves important on a manifold?

Integral curves on a manifold are important because they allow us to understand the behavior of a vector field on the manifold. They can help us visualize the direction and magnitude of the vector field at each point, and they provide a way to solve differential equations on the manifold.

How are integral curves calculated on a manifold?

The calculation of integral curves on a manifold involves solving a set of differential equations known as the Euler-Lagrange equations. These equations describe the relationship between the vector field and the coordinates on the manifold, and they allow us to determine the path of the integral curves.

What are some applications of studying integral curves on a manifold?

Studying integral curves on a manifold has applications in many fields, including physics, engineering, and computer graphics. For example, in physics, integral curves can be used to study the motion of particles in a gravitational field, while in computer graphics, they can be used to create realistic animations of fluid flow or other physical phenomena.

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