Geodesic (mathematics)

In mathematics, a geodesic is a concept that generalizes the notion of a "straight line" to curved spaces. It represents the shortest path between two points in a given geometric space, such as on a surface or in a more abstract metric space. ### Key Concepts: 1. **Curved Spaces**: In Euclidean geometry (flat space), the shortest distance between two points is a straight line.

Alexandrov's uniqueness theorem is a fundamental result in the theory of geometric measure and Riemannian geometry, particularly concerning the uniqueness of hyperbolic metrics in certain settings. Named after the Russian mathematician P.S. Alexandrov, the theorem primarily deals with the properties of spaces with non-positive curvature.

In the context of differential geometry and complex analysis, a **complex geodesic** typically refers to a generalization of the concept of a geodesic in the realm of complex manifolds or complex spaces. The classical notion of a geodesic is a curve that is locally a distance minimizer between points in a given space. In Riemannian geometry, geodesics are trajectories that exhibit extremal properties (typically, minimizing lengths) in a curved space.

Geodesic bicombing is a concept from differential geometry and metric geometry that involves defining a systematic way to describe the distances and paths (geodesics) between points in a metric space. This idea is particularly useful in the study of spaces that may not have a linear structure or may be located in more abstract settings, such as manifolds or CAT(0) spaces.

A geodesic circle is a concept in differential geometry, particularly in the study of Riemannian manifolds. It refers to the set of points that are a fixed distance (radius) from a given point on the manifold, along the shortest path, or geodesics, which are the generalization of straight lines in curved spaces.

Geodesic convexity is a concept that arises in the context of Riemannian geometry and more generally in the study of metric spaces. A set is termed geodesically convex if, for any two points within the set, the shortest path (geodesic) connecting these two points lies entirely within the set.

Geodesic curvature is a concept from differential geometry that pertains to the curvature of curves on surfaces. More specifically, it measures how much a given curve deviates from being a geodesic on a surface. To understand geodesic curvature, it's helpful to first define some basic terms: 1. **Geodesic**: A geodesic is the shortest path between two points on a surface. On a flat surface, geodesics are straight lines.

Geodesic deviation refers to the phenomenon in general relativity that describes how nearby geodesics—paths followed by free-falling particles—diverge or converge due to the curvature of spacetime. In a curved spacetime, even if an object starts out on a geodesic (which is the generalization of a straight line in curved space), the path of that object may not remain parallel to the path of a nearby object over time.

Geodesics as Hamiltonian flows refer to the representation of geodesic motion (the shortest path between points on a manifold) in the language of Hamiltonian mechanics, a framework in classical mechanics that describes the evolution of dynamical systems. ### Background Concepts 1. **Geodesics**: In differential geometry, a geodesic on a manifold is a curve that represents, locally, the shortest path between points.

In general relativity, geodesics are the paths that objects follow when they move through spacetime without any external forces acting upon them. The concept is an extension of the idea of straight lines in Euclidean geometry to the curved spacetime of general relativity. ### Key Points about Geodesics in General Relativity: 1. **Spacetime Curvature**: General relativity posits that gravity is not just a force but a curvature of spacetime caused by mass and energy.

The theorem of the three geodesics is a result in the field of differential geometry, particularly in the study of geodesics on Riemannian manifolds.