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I have a question in Manfredo do Carmo's book: Differential geometry of curves and surfaces.

According to the explanation of definition of regular surfaces, the condition 2 can avoid some kind of "self-intersection" which are shown with a figure in the book as the following:

enter image description here

However, at any point except those on the line made by the self-intersection, the surface is locally satisfies the definition of regular surface. At any point $p$ on that line, We can define two different differentiable maps x$_1$, x$_2$ which can parametrize the two pieces of self-intersection respectively. Because the definition of regular surface allows more than one parametrization x for the same point on the surface. It makes we can conclude the above figure is still regular.

Therefore, what kind situation of self-intersection of a surface does the condition 2 want to avoid?

I can't understand it by myself.

H.J. Chou
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1 Answers1

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One of the hypotheses of Manfredo's definition of regular surface is that there is an open set $U\subset \mathbb R ^ 2$ over the neighborhood $V \cap S \in \mathbb R ^ 3$. To suppose the existence of this open in the problematic line is the absurdity we seek. Remember that for a surface to be regular, it must be regular in all its points. An exercise similar to your counterexample is 10 in section 2.2 of the book.

Noob mathematician
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Marta Oliveira
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