Gauss map

In mathematics, the Gauss map for a surface ${\displaystyle S}$ lying in 3-space, associates to any point ${\displaystyle p}$ of ${\displaystyle S}$ the unit (normal) vector ${\displaystyle N=N(p)}$ that is orthogonal to the tangent plane to ${\displaystyle S}$ at ${\displaystyle p}$. Hence ${\displaystyle N}$ is a map from ${\displaystyle S}$ to the unit sphere ${\displaystyle S^{2}}$.

There arises the question of choosing the normal vector ${\displaystyle N(p)}$ among the two possible choices (namely ${\displaystyle N}$ and ${\displaystyle -N}$). Moreover we will require the Gauss map to be continuous: no jumps are allowed. There is no a priori "better" choice, however the ability to choose a (continuous) Gauss map is equivalent to the surface being orientable. However it is always locally true (i.e. on a small piece of the surface).

The notion of Gauss map can be generalized to a submanifold ${\displaystyle S}$ of dimension ${\displaystyle k}$ in an ambient manifold ${\displaystyle M}$ of dimension ${\displaystyle n}$. In that case, the unit normal vector is replaced by the tangent ${\displaystyle k}$-plane at the point ${\displaystyle p\in S}$. (It should be noted that in euclidean 3-space, a 2-plane is characterized by its unit normal vector -- up to sign, hence the definition above.) The Gauss map then goes from ${\displaystyle S}$ to the set of tangent ${\displaystyle k}$-planes in the tangent bundle ${\displaystyle TM}$. The situation is somewhat simpler in euclidean ${\displaystyle n}$-space, where the target set is the Grassmannian ${\displaystyle G_{k,n}}$: the set of all (oriented) ${\displaystyle k}$-planes in ${\displaystyle \mathbb {R} ^{n}}$. (In the introductory example, ${\displaystyle G_{2,3}\simeq G_{1,3}\simeq S^{2}}$.) In a more general manifold ${\displaystyle M}$, the target space for the Gauss map ${\displaystyle N}$ is a Grassmann bundle built on the tangent bundle ${\displaystyle TM}$. The question of the (global) existence of ${\displaystyle N}$ is then a non trivial topological question.