The topology of symmetric, second-order tensor fields

Abstract

We study the topology of symmetric, second-order tensor fields. The goal is to represent their complex structure by a simple set of carefully chosen points and lines analogous to vector field topology. We extract topological skeletons of the eigenvector fields, and we track their evolution over time. We study tensor topological transitions and correlate tensor and vector data. The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. Degenerate points play a similar role as critical points in vector fields. We identify two kinds of elementary degenerate points, which we call wedges and trisectors. They can combine to form more familiar singularities-such as saddles, nodes, centers, or foci. However, these are generally unstable structures in tensor fields. Finally, we show a topological rule that puts a constraint on the topology of tensor fields defined across surfaces, extending to tensor fields the Poincare-Hopf theorem for vector fields.<>