Smooth maps between manifolds domain restriction

In summary: Hence, we restrict the domain of ##\psi \circ F \circ \phi^{-1}## to ##\phi(U \cap F^{-1}(V))##, which guarantees that all points in the domain have well-defined images in the codomain.
  • #1
center o bass
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2
Let ##M## and ##N## be smooth manifolds and let ##F:M \to N## be a smooth map. Iff ##(U,\phi)## is a chart on ##M## and ##(V,\psi)## is a chart on ##N## then the coordinate representation of ##F## is given by ##\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \to \psi(V)##. My question is: why one restricts the domain of ##\psi \circ F \circ \phi^{-1}## to ##\phi(U \cap F^{-1}(V))## and not just ##\phi(U)##? I see one can run the risk that ##F(U) \cap V = \varnothing## and that ##\psi(\varnothing)## is not well defined. Is this the reason for the restriction on the domain?
 
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  • #2
center o bass said:
Let ##M## and ##N## be smooth manifolds and let ##F:M \to N## be a smooth map. Iff ##(U,\phi)## is a chart on ##M## and ##(V,\psi)## is a chart on ##N## then the coordinate representation of ##F## is given by ##\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \to \psi(V)##. My question is: why one restricts the domain of ##\psi \circ F \circ \phi^{-1}## to ##\phi(U \cap F^{-1}(V))## and not just ##\phi(U)##? I see one can run the risk that ##F(U) \cap V = \varnothing## and that ##\psi(\varnothing)## is not well defined. Is this the reason for the restriction on the domain?
I think what you're trying to say is right.

The basic idea is that your chart homeomorphism ##\psi## is defined only on ##V##. Thus, we have to start with elements of ##\phi(U)## (the domain of ##\phi^{-1}##) that have images that ##F## can map into ##V##. Otherwise, we could have a point ##u\in U## such that ##F(u)\not\in V##, so ##\psi(u)## is undefined.
 

What is a smooth map between manifolds?

A smooth map between manifolds is a function that maps points from one manifold (a topological space that locally resembles Euclidean space) to another while preserving certain properties, such as continuity and differentiability.

What is the significance of domain restriction in smooth maps between manifolds?

Domain restriction in smooth maps between manifolds refers to limiting the domain (input) of the function to a specific subset of the manifold. This allows for a more precise and well-defined mapping between manifolds.

What are the requirements for a smooth map to be considered valid between manifolds?

A smooth map between manifolds must be continuous, differentiable, and preserve the structure of the manifolds, meaning that points that are close to each other in the domain should also be close in the codomain (output).

What are some examples of smooth maps between manifolds?

Some examples of smooth maps between manifolds include the stereographic projection (mapping points on a sphere to points on a plane), the logarithmic map (mapping points on a manifold to points in Euclidean space), and the exponential map (mapping tangent vectors to points on a manifold).

How are smooth maps between manifolds useful in scientific research?

Smooth maps between manifolds are useful in various scientific fields, such as physics, engineering, and computer science. They allow for the representation and analysis of complex systems and can be used to model physical phenomena, optimize designs, and develop efficient algorithms.

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