Question on judging a regular surface in differential geometry
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?
condition 1 in the proposition is that x is differentiable
condition 2 is that locally homeomorphism
condition 3 is that differential dx is one-to-one
differential-geometry surfaces parametrization
asked Nov 14 at 6:56
Jaeyoon Yoo
876
add a comment |
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?
condition 1 in the proposition is that x is differentiable
condition 2 is that locally homeomorphism
condition 3 is that differential dx is one-to-one
differential-geometry surfaces parametrization
asked Nov 14 at 6:56
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?
condition 1 in the proposition is that x is differentiable
condition 2 is that locally homeomorphism
condition 3 is that differential dx is one-to-one
differential-geometry surfaces parametrization
asked Nov 14 at 6:56
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?
condition 1 in the proposition is that x is differentiable
condition 2 is that locally homeomorphism
condition 3 is that differential dx is one-to-one
differential-geometry surfaces parametrization
differential-geometry surfaces parametrization
asked Nov 14 at 6:56
Jaeyoon Yoo
876
asked Nov 14 at 6:56
Jaeyoon Yoo
876
edited Nov 21 at 5:25
asked Nov 14 at 6:56
Jaeyoon Yoo
876
asked Nov 14 at 6:56
Jaeyoon Yoo
876
asked Nov 14 at 6:56
Jaeyoon Yoo
876
876
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2997899%2fquestion-on-judging-a-regular-surface-in-differential-geometry%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
