Question on proposition 4 in do carmo's differential geometry page 66
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This is the proposition and proof saying that if we know that S is a regular surface and x is a candidate of parametrization which satisfies 1)differentiable 2)surjective differential map 3)continuous bijective then, x has a continuous inverse thus resulting that x is a real parametrization
However, in the proof, I cannot see that where I used the condition that S is a regular surface
Please let me know where the condition S is a regular surface used and is there any counter-example that x does not have a continuous inverse when S is not a regular surface?
differential-geometry parametrization
asked 15 hours ago
Jaeyoon Yoo
876
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up vote
1
down vote
favorite
This is the proposition and proof saying that if we know that S is a regular surface and x is a candidate of parametrization which satisfies 1)differentiable 2)surjective differential map 3)continuous bijective then, x has a continuous inverse thus resulting that x is a real parametrization
However, in the proof, I cannot see that where I used the condition that S is a regular surface
Please let me know where the condition S is a regular surface used and is there any counter-example that x does not have a continuous inverse when S is not a regular surface?
differential-geometry parametrization
asked 15 hours ago
Jaeyoon Yoo
876
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
This is the proposition and proof saying that if we know that S is a regular surface and x is a candidate of parametrization which satisfies 1)differentiable 2)surjective differential map 3)continuous bijective then, x has a continuous inverse thus resulting that x is a real parametrization
However, in the proof, I cannot see that where I used the condition that S is a regular surface
Please let me know where the condition S is a regular surface used and is there any counter-example that x does not have a continuous inverse when S is not a regular surface?
differential-geometry parametrization
asked 15 hours ago
Jaeyoon Yoo
876
This is the proposition and proof saying that if we know that S is a regular surface and x is a candidate of parametrization which satisfies 1)differentiable 2)surjective differential map 3)continuous bijective then, x has a continuous inverse thus resulting that x is a real parametrization
However, in the proof, I cannot see that where I used the condition that S is a regular surface
Please let me know where the condition S is a regular surface used and is there any counter-example that x does not have a continuous inverse when S is not a regular surface?
differential-geometry parametrization
differential-geometry parametrization
asked 15 hours ago
Jaeyoon Yoo
876
asked 15 hours ago
Jaeyoon Yoo
876
asked 15 hours ago
Jaeyoon Yoo
876
asked 15 hours ago
Jaeyoon Yoo
876
asked 15 hours ago
Jaeyoon Yoo
876
876
add a comment |
add a comment |
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