Riemannian Geometry for Scientific Visualization

Abstract

This tutorial introduces the most important basics of Riemannian geometry and related concepts with a specific focus on applications in scientific visualization. The main concept in Riemannian geometry is the presence of a Riemannian metric on a differentiable manifold, comprising a second-order tensor field that defines an inner product in each tangent space that varies smoothly from point to point. Technically, the metric is what allows defining and computing distances and angles in a coordinate-independent manner. However, even more importantly, it in a sense is really the major structure (on top of topological considerations) that defines the space where scientific data, such as scalar, vector, and tensor fields live.