Local flow as a diffeomorphism.

In summary, on a smooth manifold, a smooth vector field V guarantees the existence of a unique integral curve φ^(p): J → M for some open interval J ⊆ ℝ, where 0 ∈ J and φ^(p)(0) = V_(φ^(p)(0)). This allows for the definition of a map φ_t: φ^(p)(J) → M such that φ_t(p) = φ^(p)(t). The map satisfies the properties φ_t ∘ φ_s = φ_(t+s) and φ_0(p) = p, and is often stated as a diffeomorphism. To prove this, one must show that the map has a differentiable inverse and determine the subsets of M that
  • #1
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Given a smooth vector field ##V## on a smooth manifold ##M## the uniqueness of differential equations assures
that there exists a unique integral curve ##\phi^{(p)}: J \to M## for some open interval ##J \subseteq \mathbb{R}## for which ##0 \in J## and ##\dot \phi^{(p)} (0) = V_{\phi^{(p)} (0)}##. We can now define a map ##\phi_t: \phi^{(p)}(J) \to M## such that ##\phi_t(p) = \phi^{(p)}(t)##. From the fact that we can reparametrize the above solution it follows that we have the properties ##\phi_t \circ \phi_s = \phi_{t+s}##. As an application of the chain rule it's easy to show that this map satisfies ##\phi_t \circ \phi_s = \phi_{t+s}## and it obviously satisfies ##\phi_0(p) = p##. However it's also often stated that it is a diffeomorphism. How does one go about proving that? Sure the map is differentiable, but how does one prove it has an differentiable inverse? Furthermore exactly what subsets of ##M## are the domain and codomains of the diffeomorphism?
 
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  • #2
Notice that φtφ-t = φ0 = φ-tφt which shows the diffeomorphism property. In general, the second question is complicated (and depends heavily on your choice of manifold), but in many special cases it turns out to be the whole manifold.
 

What is local flow as a diffeomorphism?

Local flow as a diffeomorphism refers to a mathematical concept in differential topology and dynamical systems. It involves the study of continuous transformations that preserve the local structure of a manifold, known as diffeomorphisms.

How is local flow as a diffeomorphism different from global flow?

Global flow refers to continuous transformations that preserve the entire structure of a manifold, while local flow only preserves the structure in a small neighborhood of a point. This means that local flow can have different effects in different regions of the manifold, while global flow has a consistent effect throughout.

Why is local flow as a diffeomorphism important?

Local flow as a diffeomorphism is important in understanding the behavior of dynamical systems and studying the local properties of manifolds. It also has applications in physics, engineering, and other fields where continuous transformations are relevant.

What is a diffeomorphism group?

A diffeomorphism group is a set of all diffeomorphisms on a given manifold, together with a composition operation. It forms a mathematical group, meaning that it satisfies the properties of closure, associativity, identity, and invertibility.

How is local flow as a diffeomorphism used in practical applications?

Local flow as a diffeomorphism has various practical applications, such as in computer graphics, image processing, and fluid dynamics. It is also used in the study of dynamical systems, chaos theory, and differential equations. Additionally, it has applications in shape analysis and pattern recognition.

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