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I was reading differential geometry from a lecture note, there I found the following theorem,

If $S$ is a compact connected oriented $n-$ surface in $\mathbb{R}^{n+1}.$ Then the gauss map is surjective.

Now in the proof , the picture of the surface $S$ sketched as if $S$ is homeomorphic to $S^{n-1}$ or$D^{n}.$

I am wondering that, is it true?

Is there any characterisation of $n-$ surfaces?

Definition: An $n-$ surface $S$ is a level curve of a smooth function whose gradient is non-zero on $S.$

SOUL
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1 Answers1

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No. Consider the following function: $$ f(x,y,z)=(x^2+y^2-1)^2+z^2 $$ Note that $f^{-1}(\{0\})$ is the unit circle in the $xy$-plane, while $f^{-1}(\epsilon)$ is a torus around that circle for sufficiently small $\epsilon>0$. For instance, the $\epsilon=1/2$ case looks like this:

plot of level surface from wolframalpha.com

One can in fact obtain any oriented closed surface as the regular level set of a smooth function (see here for some examples).

As an aside, the unit disc (open or closed) is not applicable to this statement (since it concerns compact surfaces without boundary), and indeed the Gauss map of a disc need not be surjective.

Kajelad
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  • . . . nice! . . – janmarqz May 24 '21 at 17:51
  • @Kajelad can you plot the graph of the function for small $\epsilon?$ – SOUL May 24 '21 at 18:09
  • @Kajelad another question, is regular level set of some smooth function in $\mathbb{R}^{n+1}$ a manifold? – SOUL May 24 '21 at 18:18
  • @Tom. Yes, this is a special case of the preimage theorem. Such a level set is always a smooth, orientable, closed, codimension one submanifold, but it won't necessarily be compact or connected. – Kajelad May 24 '21 at 18:43
  • @Kajelad Is there any general equations of torus in $\mathbb{R}^3?$ – SOUL May 25 '21 at 02:12
  • @Tom. I don't know what you mean by that. – Kajelad May 25 '21 at 03:07
  • For example $x^2+y^2+2ax+2by+c=0$ represents all circles in $\mathbb{R}^2 .$ So, I am wondering, is there any for torus?@Kajelad – SOUL May 25 '21 at 04:57
  • Also, can you please suggest me a good introductory level book on differential geometry which will help me to get the big picture? @Kajelad – SOUL May 25 '21 at 04:58
  • @Tom. That equation describes a geometric notion of a circle, but if you're talking about topological circles (i.e. regular curves homeomorphic to the circle), then there are many other curves which are not described by such a nice equation. The same is true of Tori in $\mathbb{R}^3$. If you want all surfaces homeomorphic to a torus, then there are too many possibilities to describe with a finite set of parameters. – Kajelad May 25 '21 at 05:30
  • @Tom. The "standard" text for this kind of thing is Do Carmo's Differential Geometry of Curves and Surfaces. This question contains many other recommendations, which might be of interest. – Kajelad May 25 '21 at 05:32
  • @Kajelad, Another question and final question, if we consider all those surfaces generated by revolving one circle touching another circle and revolving about an axis passing through its centre, then we can get a class of torus, right? – SOUL May 25 '21 at 06:24
  • @Kajelad you have also mentioned that any orientable regular surface can be obtained as level set of a smooth function, can you give me a reference for that? – SOUL May 25 '21 at 06:25
  • @Tom. Surfaces of revolution of a circle would certainly give a class of tori. I don't know of a good reference for the fact that any closed (i.e. properly embedded) orientable surface is a level set, and, as far as I can tell, it requires some "high-powered" tools to prove; namely Alexander duality and partitions of unity. – Kajelad May 25 '21 at 14:55