- #1
spaghetti3451
- 1,344
- 33
If ##M^{2} \subset \mathbb{R}^{3}## is a surface with given normal field, we define the Gauss (normal) map
$$n:M^{2} \rightarrow \text{unit sphere}\ S^{2}$$
by
$$n(p) = \textbf{N}(p), \qquad \text{the unit normal to $M$ at $p$}.$$
-------------------------------------------------------------------------------------------------------------------------------------------------------
It can be shown that
$$n^{*}\text{vol}^{2}_{S}=K\text{vol}^{2}_{M},$$
where ##\text{vol}^{2}_{S}## is the volume form in ##S^{2}##, ##\text{vol}^{2}_{M}## is the volume form of ##M^{2}## and ##K## is the Gauss curvature.
This means that
1. the Gauss map is a local diffeomorphism in the neighbourhood ##U## of any ##p \in M^{2}## at which ##K(p)\neq 0## (alternatively, ##u \in M## is a regular point for the Gauss map provided ##K(u)\neq 0##)
2. if ##U## is positively oriented then ##n(U)## will be positively oriented on ##S^2## iff ##K>0##.
-------------------------------------------------------------------------------------------------------------------------------------------------------
Let ##y \in V## be a regular value of ##\phi: M^{n} \rightarrow V^{n}##; that is, ##\phi_{*}## at ##\phi^{-1}(y)## is onto. For each ##x \in \phi^{-1}(y)##, ##\phi_{*}:M_{x}\rightarrow V_{y}## is also ##1:1##; that is, ##\phi_{*}## is an isomorphism. Put
$$\text{sign}\ \phi(x) := \pm 1$$
where the ##+## sign is used iff ##\phi_{*}:M_{x}\rightarrow V_{y}## is orientation-preserving. Then
$$\text{deg}(\phi)=\sum_{x \in \phi^{-1}(y)} \text{sign}\ \phi(x)$$
-------------------------------------------------------------------------------------------------------------------------------------------------------
1. Why does ##n^{*}\text{vol}^{2}_{S}=K\text{vol}^{2}_{M}## mean that the Gauss map is a local diffeomorphism in the neighbourhood ##U## of any ##p \in M^{2}## at which ##K(p)\neq 0##?
2. Why does ##n^{*}\text{vol}^{2}_{S}=K\text{vol}^{2}_{M}## mean that ##u \in M## is a regular point for the Gauss map provided ##K(u)\neq 0##?
3. Why does ##n^{*}\text{vol}^{2}_{S}=K\text{vol}^{2}_{M}## mean that, if ##U## is positively oriented then ##n(U)## will be positively oriented on ##S^2## iff ##K>0##?
4. How can we use these three facts to evaluate the Brouwer degree of the Gauss normal map?
$$n:M^{2} \rightarrow \text{unit sphere}\ S^{2}$$
by
$$n(p) = \textbf{N}(p), \qquad \text{the unit normal to $M$ at $p$}.$$
-------------------------------------------------------------------------------------------------------------------------------------------------------
It can be shown that
$$n^{*}\text{vol}^{2}_{S}=K\text{vol}^{2}_{M},$$
where ##\text{vol}^{2}_{S}## is the volume form in ##S^{2}##, ##\text{vol}^{2}_{M}## is the volume form of ##M^{2}## and ##K## is the Gauss curvature.
This means that
1. the Gauss map is a local diffeomorphism in the neighbourhood ##U## of any ##p \in M^{2}## at which ##K(p)\neq 0## (alternatively, ##u \in M## is a regular point for the Gauss map provided ##K(u)\neq 0##)
2. if ##U## is positively oriented then ##n(U)## will be positively oriented on ##S^2## iff ##K>0##.
-------------------------------------------------------------------------------------------------------------------------------------------------------
Let ##y \in V## be a regular value of ##\phi: M^{n} \rightarrow V^{n}##; that is, ##\phi_{*}## at ##\phi^{-1}(y)## is onto. For each ##x \in \phi^{-1}(y)##, ##\phi_{*}:M_{x}\rightarrow V_{y}## is also ##1:1##; that is, ##\phi_{*}## is an isomorphism. Put
$$\text{sign}\ \phi(x) := \pm 1$$
where the ##+## sign is used iff ##\phi_{*}:M_{x}\rightarrow V_{y}## is orientation-preserving. Then
$$\text{deg}(\phi)=\sum_{x \in \phi^{-1}(y)} \text{sign}\ \phi(x)$$
-------------------------------------------------------------------------------------------------------------------------------------------------------
1. Why does ##n^{*}\text{vol}^{2}_{S}=K\text{vol}^{2}_{M}## mean that the Gauss map is a local diffeomorphism in the neighbourhood ##U## of any ##p \in M^{2}## at which ##K(p)\neq 0##?
2. Why does ##n^{*}\text{vol}^{2}_{S}=K\text{vol}^{2}_{M}## mean that ##u \in M## is a regular point for the Gauss map provided ##K(u)\neq 0##?
3. Why does ##n^{*}\text{vol}^{2}_{S}=K\text{vol}^{2}_{M}## mean that, if ##U## is positively oriented then ##n(U)## will be positively oriented on ##S^2## iff ##K>0##?
4. How can we use these three facts to evaluate the Brouwer degree of the Gauss normal map?