visualizing singularities of maps from sphere to R^2 - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:19:46Z http://mathoverflow.net/feeds/question/116808 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116808/visualizing-singularities-of-maps-from-sphere-to-r2 visualizing singularities of maps from sphere to R^2 John Mangual 2012-12-19T18:23:33Z 2012-12-19T23:42:34Z <p>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:</p> <p>$\left[\begin{array}{cc}\frac{\partial f_1}{\partial x} &amp; \frac{\partial f_1}{\partial y}\\ \frac{\partial f_2}{\partial x} &amp; \frac{\partial f_2}{\partial y} \end{array} \right]$</p> <p>has less than full rank. Locally, can we draw a picture of what singularities look like?</p> <p>In the case of maps $S^2 \to \mathbb{R}^1$, we just get the critical points where $f(x,y) = f(x_0,y_0) + (x-x_0,y-y_0)^T (D^2 f )(x-x_0,y-y_0)$</p> http://mathoverflow.net/questions/116808/visualizing-singularities-of-maps-from-sphere-to-r2/116811#116811 Answer by Robert Bryant for visualizing singularities of maps from sphere to R^2 Robert Bryant 2012-12-19T19:41:16Z 2012-12-19T23:42:34Z <p>I think you want to look at Guillemin and Golubitsky's book <em>Stable mappings and their singularities</em>, which has a thorough description of what the singularity types of stable mappings are between surfaces.</p> <p>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.</p> <p>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.</p>