Geodesics

Definition

Geodesics are the shortest paths between points on a curved surface or in a curved space, analogous to straight lines in Euclidean geometry. They play a vital role in understanding the structure of spaces, especially in relation to curvature and convexity, as they help define the intrinsic geometry of surfaces. Geodesics reveal how distances and angles behave differently than in flat spaces, highlighting the unique properties of curved surfaces.

5 Must Know Facts For Your Next Test

1. On a convex surface, geodesics are always locally distance-minimizing curves, meaning they represent the shortest distance between points along the surface.
2. The existence and uniqueness of geodesics can be guaranteed under certain conditions related to curvature, such as in Alexandrov's theorem for convex surfaces.
3. Geodesics can exhibit interesting behaviors on different surfaces; for example, on a sphere, they are represented by great circles.
4. The study of geodesics helps in understanding the global geometry of a surface, including how curves can loop back on themselves or diverge.
5. In general relativity, geodesics describe the motion of objects under the influence of gravity, illustrating how spacetime curvature affects trajectories.

Review Questions

• How do geodesics relate to the concepts of curvature and convexity on surfaces?
  • Geodesics are directly influenced by the curvature of a surface; on convex surfaces, they represent the shortest paths between points. The curvature determines how these paths behave: in positively curved spaces like spheres, geodesics can curve outward and meet again, while in negatively curved spaces, they may diverge. Thus, understanding geodesics helps in grasping how curvature impacts the overall geometry and distances on a surface.
• Discuss Alexandrov's theorem and its implications for geodesics on convex surfaces.
  • Alexandrov's theorem states that any two points on a complete convex surface can be connected by a unique geodesic that is locally distance-minimizing. This theorem implies that convex surfaces have well-defined properties regarding their geodesics, ensuring that they behave predictably and consistently. It also leads to conclusions about the global structure of such surfaces, allowing mathematicians to classify them based on their geodesic behavior.
• Evaluate the significance of studying geodesics in Riemannian geometry and its applications in other fields.
  • Studying geodesics in Riemannian geometry is crucial because it helps reveal fundamental properties about curved spaces and their intrinsic geometries. Applications extend beyond pure mathematics; for instance, in physics, geodesics describe how objects move under gravitational influence according to general relativity. This connection between geometry and physics illustrates how understanding geodesics enhances our grasp of the universe’s structure and behavior.

Related terms

Curvature:

A measure of how much a geometric object deviates from being flat or straight, often quantified in terms of Gaussian curvature for surfaces.

Convexity:

A property of a set or surface where any two points within it can be connected by a line segment that lies entirely within that set or surface.

Riemannian Geometry: A branch of differential geometry that studies smooth manifolds with a Riemannian metric, allowing for the definition of geodesics on curved surfaces.