Gauss map for an $n$-manifold in $\mathbb{R}^{n+1}$

Question

If $M$ is an embedded $2$-manifold in $\mathbb{R}^3$ then the gauss map in local coordinates is given by:

$\theta(u,v)=\frac{\frac{\partial\phi}{\partial u}\times\frac{\partial\phi}{\partial v}}{\lVert\frac{\partial\phi}{\partial u}\times\frac{\partial\phi}{\partial v}\rVert}\ldots (i)$

where $\phi:U\subset\mathbb{R}^2\rightarrow M$ is a local parametrization around some point in $M$ and $\theta:\phi(U)\rightarrow S^2$ is the gauss map. The explicit formula $(i)$ tells us why the gauss map can be defined globally iff $M$ is orientable.

So my question is whether there is a similar explicit local formula for the gauss map if $M$ is an $n$-manifold in $\mathbb{R}^{n+1}$? Will it follow similarly as in the case above that the gauss map can be defined globally iff $M$ is orientable?

Answer 1

Yes, the Gauss map in local coordinates is given by $$\theta(u_1,...,u_n)=\frac{\frac{\partial\phi}{\partial u_1}\times...\times \frac{\partial\phi}{\partial u_n}}{\lVert\frac{\partial\phi}{\partial u_1}\times...\times \frac{\partial\phi}{\partial u_n}\rVert}$$

where the generalized cross product of $v_1,...,v_n\in\mathbb R^{n+1}$ is defined to be the formal determinant

$$ v_1\times...\times v_n=\det\begin{vmatrix}&v_1&\\&\vdots&\\&v_n&\\e_1&\cdots&e_{n+1}\end{vmatrix} $$ That the Gauss map can be defined globally iff the manifold is orientable follows similarly, since one property of the generalized cross product is that for $v_1,...,v_n$ linear independent $(v_1,...,v_n,v_1\times...\times v_n)$ is positively oriented.