# A Converse to the Gauss Bonnet Theorem

Let $S$ be a compact surface in $\mathbb{R}^{3}$ with the gauss normal map $N:S\to \mathbb{S}^{2}$. Assme that $\phi;\mathbb{S}^{2}\to S$ is a diffeomorphism. Put $F=N\circ \phi$ and represent $F:\mathbb{S}^{2}\to \mathbb{S}^{2}$ in the form $F=(f,g,h)$. then as a consequence of the Gauss Bonnet theorem we have $$\iint_{\mathbb{S^{2}}} fdgdh= 4/3 \; \pi$$ (See page 14 of the(printed version) of the book, Non commutative geometry by Alain Connes

My question is that ;is the converse of the above statment true?

That is:

Assume that $F:\mathbb{S}^{2}\to \mathbb{S}^{2}$ is a smooth map with $F=(f,g,h)$ such that the above integral equality hold. Is there a compact surface $S$ with Gauss normal map $N$ and a diffeomorphism $\phi: \mathbb{S}^{2} \to S$ such that $F=N\circ \phi$?

-
What about $N=\operatorname{id}$ and $\phi=F$? – Alex Degtyarev Feb 12 '14 at 18:33

$\newcommand{\bR}{\mathbb{R}}$ If $\Sigma\subset \bR^3$ is a cooriented surface then its Gauss map $\Gamma:\Sigma\to S^2$ has a symplectic nature. Its graph, viewed as a submanifold of $\bR^3\times S^2$ is a Legendrian submanifold with respect to the canonical contact structure on $\bR^3\times S^2$.

The Legendrian condition imposes restrictions on the types of singularities of this map. In other words, there are also local obstructions to the converse.

This is similar with the fact that a smooth map $F:\bR^n\to\bR^n$ need not be the differential of a function $U:\bR^n\to\bR$. If it where, the graph of $F$ would be a lagrangian submanifold of $\bR^{2n}$ with the canonical symplectic structure. For details see Volume 1 of

Arnold, Gusein-Zade, Varchenko : Singularities of Differentiable Maps.

-
thank you for your interesting answer and useful information. – Ali Taghavi Feb 14 '14 at 14:03
But I have some questions:what is the restriction on singularities? Do these restriction invariant after diffeo $\phi$? Is there a degree-1 map on $\mathbb{S}^{2}$ which singularities does not satisfy the restriction- condition? – Ali Taghavi Feb 14 '14 at 14:07
The answer to your questions is rather involved. The book I mentioned has many more details. The nature of the singularities is independent of the choice of local coordinates and in particular independent of the diffeomorphism $\phi$. – Liviu Nicolaescu Feb 15 '14 at 13:26

First of all, the identity holds for any degree 1 map $F:\mathbb S^2\to\mathbb S^2$. Moreover, for any $F=(f,g,h):\mathbb S^2\to\mathbb S^2$, $$\int_{\mathbb S^2} f\,dgdh = \frac43\pi \deg F.$$ This follows from Stokes' formula. So basically the identity means that the Gauss map of a surface diffeomorphic to the sphere has degree 1. This is by the way not always true, you have to choose $N$ according to the orientation (that you use to define the degree), otherwise you get the value $-\frac43\pi$. In other words, $\phi$ must be orientation-preserving.

Back to the question, the answer is trivially yes if $F$ is a diffeomorphism, by Alex Degtyarev's comment. In general, the answer is no because certain types of singularities can not occur.

For example, consider $F$ that maps some circle $C\subset\mathbb S^2$ to the north pole but such that the $F$-image of an open disc $D\subset\mathbb S^2$ bounded by $C$ avoids the north and south poles. It is easy to construct such a map of degree 1. Suppose that the desired $S$ and $\phi$ exist and consider the disc $\phi(D)$ in $S$. On its boundary $\phi(C)$ the normal is vertical, hence $\phi(C)$ is contained in some horizontal plane $H$. Then there is a point in the interior of $\phi(D)$ where the tangent plane is horizontal (look at a point furthest from $H$). The normal at this point is vertical contrary to the fact that $F(D)$ avoids the north and south poles.

-
thank you very much for your interesting answer. why there is an interior point which tangent space is horizontal? – Ali Taghavi Feb 14 '14 at 14:17
Take a point with the highest $z$-coordinate. If it is on the boundary, take one with the lowest $z$-coordinate. Since the entire boundary is on the same level, either a highest or lowest point is in the interior. – Sergei Ivanov Feb 14 '14 at 16:27
thanks for the comment. what is a degree-1 map with the property which you mentioned? – Ali Taghavi Feb 15 '14 at 9:56
For example, contract a hemisphere to a segment by contracting each radial circle to a point. The resulting space is a bouquet of a sphere and a circle. Place the segment to the sphere by an injective map. This can be made smooth. – Sergei Ivanov Feb 17 '14 at 16:24
Thank you very much for your very beautiful and intuitive answer. It was difficult for me to choose an answer to accept. I hope that this restriction in MO would be removed:meta.mathoverflow.net/questions/1491/… – Ali Taghavi Feb 23 '14 at 13:44