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A question on definition of regular surface

Surface ParameterizationsProblem with definition of regular surface in classical differential geometryQuestion about focal surfaceDefinition of a regular surfaceLet $A subset S$ be a subset of a regular surface $S$. Prove that if $A$ is itself regular then $A$ is open in $S$.Find conditions for a surface of rotation to be regularRegular surface and self-intersectionsdefinition of differentiability on a regular surfaceIs surface regularity preserved under diffeomorphisms?Why do we ask for X to be open when defining a regular surface?

1

$begingroup$

I found a question in Manfredo do Carmo's famous book: Differential geometry of curves and surfaces. According to the book, it says that the condition 2 of the definition of regular surface is to avoid some kind of self-intersection and also show a figure of this situation

However, I think the surface in the figure is also a regular surface. For the self-intersection part of the surface, We can use two different differentiable maps x$_1$, x$_2$ to parametrize the two intersecting branches individually. It works because each branch is just a curl plane and the definition of regular surface allows not at most one parametrization x for each point. In sum, my question is does the figure explain the situation the condition 2 want to avoid? And furthermore, what kind of situation of self-intersection does the condition 2 want to avoid?

differential-geometry

share|cite|improve this question

edited Mar 19 at 4:37

YuiTo Cheng

2,1963937

asked Mar 19 at 4:21

DavidchouDavidchou

62

$endgroup$

add a comment | 

1

$begingroup$

I found a question in Manfredo do Carmo's famous book: Differential geometry of curves and surfaces. According to the book, it says that the condition 2 of the definition of regular surface is to avoid some kind of self-intersection and also show a figure of this situation

However, I think the surface in the figure is also a regular surface. For the self-intersection part of the surface, We can use two different differentiable maps x$_1$, x$_2$ to parametrize the two intersecting branches individually. It works because each branch is just a curl plane and the definition of regular surface allows not at most one parametrization x for each point. In sum, my question is does the figure explain the situation the condition 2 want to avoid? And furthermore, what kind of situation of self-intersection does the condition 2 want to avoid?

differential-geometry

share|cite|improve this question

edited Mar 19 at 4:37

YuiTo Cheng

2,1963937

asked Mar 19 at 4:21

DavidchouDavidchou

62

$endgroup$

add a comment | 

$begingroup$

I found a question in Manfredo do Carmo's famous book: Differential geometry of curves and surfaces. According to the book, it says that the condition 2 of the definition of regular surface is to avoid some kind of self-intersection and also show a figure of this situation

However, I think the surface in the figure is also a regular surface. For the self-intersection part of the surface, We can use two different differentiable maps x$_1$, x$_2$ to parametrize the two intersecting branches individually. It works because each branch is just a curl plane and the definition of regular surface allows not at most one parametrization x for each point. In sum, my question is does the figure explain the situation the condition 2 want to avoid? And furthermore, what kind of situation of self-intersection does the condition 2 want to avoid?

differential-geometry

share|cite|improve this question

edited Mar 19 at 4:37

YuiTo Cheng

2,1963937

asked Mar 19 at 4:21

DavidchouDavidchou

62

$endgroup$

share|cite|improve this question

edited Mar 19 at 4:37

YuiTo Cheng

2,1963937

asked Mar 19 at 4:21

DavidchouDavidchou

62

share|cite|improve this question

edited Mar 19 at 4:37

YuiTo Cheng

2,1963937

asked Mar 19 at 4:21

DavidchouDavidchou

62

edited Mar 19 at 4:37

YuiTo Cheng

2,1963937

edited Mar 19 at 4:37

YuiTo Cheng

2,1963937

asked Mar 19 at 4:21

DavidchouDavidchou

62

asked Mar 19 at 4:21

DavidchouDavidchou

62