DescriptionThis course introduces the most important basics and concepts of Riemannian geometry with a focus on applications in scientific visualization. We have two main goals: (1) Introduce Riemannian geometry to scientific visualization researchers and practitioners, and (2) Introduce researchers working in/with differential geometry or mathematical physics to important applications in scientific visualization. Fitting the colorful world theme of Siggraph Asia 2022, we target bridging the gap between the two disciplines, and connect researchers in visualization with researchers in computer graphics, geometry, physically-based animation, and continuum mechanics. The Riemannian metric is in fact the major structure that defines the space where scientific data in visualization, such as scalar, vector, and tensor fields live, which is therefore of crucial importance in scientific visualization. However, the concept of a metric, and related concepts such as affine connections and covariant derivatives, are rarely used in visualization papers. In contrast to concepts of differential topology, which have been used extensively in visualization, e.g., in scalar and vector field topology, concepts from Riemannian geometry have been under-represented in the visualization literature. One reason for this might be that most visualization techniques are developed for scalar, vector, or tensor fields given in Euclidean space, and data given on curved surfaces are usually treated explicitly through their embedding in 3D Euclidean space. However, the presence of a Riemannian metric on a manifold even has very important implications for data given in Euclidean space, for example regarding the physical meaning of visualizations as well as for the correct use of non-Cartesian coordinates. Considering the metric tensor field explicitly provides several important benefits. In this course, we will particularly highlight the additional insight that can be gained from using Riemannian geometry in scientific visualization, but we will also highlight computational advantages.