Krümmung

Krümmung, a German term meaning "curvature," refers to a fundamental concept in differential geometry that quantifies how a curve or surface deviates from being flat or straight. In mathematics, curvature describes the rate at which a geometric object bends at a given point. It is a measure of how sharply a curve or surface turns or twists as one moves along it.

For curves in a plane, curvature is defined as the magnitude of the derivative of the unit tangent vector with respect to arc length. A circle of radius *r* has constant curvature *1/r*, where smaller radii correspond to higher curvature. In higher dimensions, curvature generalizes to describe the bending of surfaces or the twisting of higher-dimensional manifolds.

In differential geometry, curvature is often expressed using the Riemann curvature tensor, which captures the intrinsic geometry of a manifold. This tensor generalizes concepts like Gaussian curvature (for surfaces) and Ricci curvature (which plays a key role in general relativity). Positive curvature implies a locally spherical shape, while negative curvature suggests saddle-like behavior.

Curvature is essential in physics, particularly in Einstein’s theory of general relativity, where the curvature of spacetime is determined by the distribution of matter and energy. It also appears in computer graphics, robotics, and materials science, where understanding bending and deformation is critical.

The study of curvature spans various fields, including Euclidean, non-Euclidean, and Riemannian geometries, each offering distinct interpretations of how space can bend and warp.