How can one generalize the Gauss map to higher dimensions? More specifically, bi-dimensional manifolds in $\mathbb{R}^4$

Question

It's easy to define the unitary tangent fields of a $2$-dimensional surface $S: I \times J \to \mathbb{R}^4$, but since I don't have the cross product anymore, an unitary normal field is harder to find.

Answer 1

One way to generalize the Gauss map is to convince yourself that in the classical setting for of surfaces in $\mathbb{R}^3$ a choice of unit normal field is equivalent to a choice of orientation, because the tangent planes determine the normal direction at each point (in the presence of a metric). If you drop the orientation, then you only have the family of tangent planes, that you can view as a mapping from your surface into the space of $2$-dimensional subspaces in $\mathbb{R}^3$. In other words, modulo the orientation, the Gauss map is determined by either tangent planes or normal lines.

Formally speaking, the range of the Gauss map (dropping the orientation) is the Grassmanian $\mathbf{Gr}(1, 3) \cong \mathbf{Gr}(2, 3) \cong \mathbb{P}^2$, that the projective plane, which locally looks like the $2$-sphere.

In your case (2-surface in $\mathbb{R}^4$), knowing locally an orthogonal frame is sufficient to think of this frame as a mapping from the surface into the Grassmanian $\mathbf{Gr}(2, 4)$.