Riemannian

Riemannian refers to concepts, structures, and mathematical ideas related to Riemannian geometry, a branch of differential geometry developed by Bernhard Riemann in the 19th century. Riemannian geometry studies smooth manifolds equipped with a Riemannian metric, a function that defines an inner product on the tangent space at each point. This metric allows for the measurement of lengths, angles, and volumes within the manifold, generalizing classical Euclidean geometry to curved spaces.

A central object in Riemannian geometry is the Riemannian manifold, a differentiable manifold endowed with a smoothly varying positive-definite metric tensor. This framework enables the definition of geodesics, which are curves that locally minimize distance, and curvature, which describes how the manifold deviates from being flat. Curvature notions such as sectional, Ricci, and scalar curvature provide insights into the geometric and topological properties of the manifold.

Riemannian geometry has wide applications in physics, particularly in general relativity, where spacetime is modeled as a four-dimensional Riemannian or pseudo-Riemannian manifold with a metric tensor describing gravitational effects. In mathematics, it is fundamental to fields like geometric analysis, topology, and global analysis.

The theory extends to various specialized topics, including Riemannian submersions, harmonic maps, and comparison theorems, which relate the geometric properties of a manifold to those of model spaces. Overall, "Riemannian" signifies a focus on the geometric structures and properties that arise from Riemannian metrics, facilitating the study of curved spaces in both theoretical and applied contexts.