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- Clarification about submanifold definition and diffeomorphisms involved in ##\mathbb R^2##

Hi,

a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of ##\mathbb R^2##.

Since it is a regular submanifold there is a global chart ##(\phi, \mathbb R^2)## such that ##\Gamma## is represented as ##(x,0),x∈\mathbb R##. Restricting such a chart to ##\Gamma## we get the manifold structure on it w.r.t. the inclusion map ##i:\Gamma ↪\mathbb R^2## is an embedding. So far so good.

My point is: is the above map ##γ:\mathbb R→\mathbb R^2## a diffeomorphism onto its image ? I believe the answer is positive.

##\Gamma## indeed is diffeomorphic to ##\mathbb R## by definition of chart. Now the composition ##g = \phi \circ \gamma## is a continuous injective map ##g: \mathbb R \rightarrow \mathbb R##. By virtue of Invariance of domain theorem ##g## is an homeomorphism hence the map ##\gamma## is actually a differentiable homeomorphism onto its image (i.e. a diffeomorphism onto the image).

a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of ##\mathbb R^2##.

Since it is a regular submanifold there is a global chart ##(\phi, \mathbb R^2)## such that ##\Gamma## is represented as ##(x,0),x∈\mathbb R##. Restricting such a chart to ##\Gamma## we get the manifold structure on it w.r.t. the inclusion map ##i:\Gamma ↪\mathbb R^2## is an embedding. So far so good.

My point is: is the above map ##γ:\mathbb R→\mathbb R^2## a diffeomorphism onto its image ? I believe the answer is positive.

##\Gamma## indeed is diffeomorphic to ##\mathbb R## by definition of chart. Now the composition ##g = \phi \circ \gamma## is a continuous injective map ##g: \mathbb R \rightarrow \mathbb R##. By virtue of Invariance of domain theorem ##g## is an homeomorphism hence the map ##\gamma## is actually a differentiable homeomorphism onto its image (i.e. a diffeomorphism onto the image).

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