Frankel 1.3 Problems


What would be wrong in defining

$$||X||^2 = \sum_i (X_U^i)^2 \,\,\, ?$$

Defining the magnitude this way would lead to a result that depends on the choice of coordinates.


Let the torus \(T^2\) be embedded in \(\mathbb{R}^3\) such that it lies flat atop the xy plane. The map \(F : T^2 \to \mathbb{R}^2\) project points on the torus to the xy plane.  By inspection, the critical points of \(F\) are those along the inner and outer circles in the middle of the torus. At these points the tangent space is perpendicular to the xy plane and therefore the differential \(F_*\) is not onto.