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Is there a classification of singularities from $S^2 \to \mathbb{R}^2$ ? The critical points of the map $(x,y) \mapsto (f_1(x,y),f_2(x,y))$ where the matrix:

\[ \left[\begin{array}{cc}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y}\\\\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{array} \right] \]

has less than full rank. Locally, can we draw a picture of what singularities look like?

In the case of maps $S^2 \to \mathbb{R}^1$, we just get the critical points where $f(x,y) = f(x_0,y_0) + (x-x_0,y-y_0)^T (D^2 f )(x-x_0,y-y_0)$

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up vote 8 down vote accepted

I think you want to look at Guillemin and Golubitsky's book Stable mappings and their singularities, which has a thorough description of what the singularity types of stable mappings are between surfaces.

Basically, the only stable singularities for smooth maps between surfaces (i.e., $2$-manifolds) are folds and cusps, and these cannot usually perturbed away by small perturbations.

If you don't restrict to stable mappings or some similar class, the kinds of singularities that can occur can be extremely complicated, and I doubt that there is any workable classification.

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