geometry of gradients and second derivatives

Building on conversations we had yesterday about the geometry and topology of gradients of a scalar field, Gaby Contardo (Flatiron) and I worked out at the end of the day today that valleys of a density field (meaning here a many-times differentiable smooth density model in some d-dimensional space) can be traced by looking for paths along which the density gradient has zero projection onto the principal component (largest-eigenvalue eigenvector) of the second-derivative tensor (the Hessian, to some). We looked at some toy-data examples and this does look promising as a technique for tracing or finding gaps or low-density regions in d-dimensional point clouds.