Parallel transport on a triangle on a sphere refers to the process of moving a vector or geometric object along a path on the surface of a sphere, while maintaining its direction and orientation relative to the surface. It is commonly used in geometry and physics to study properties of curved spaces.
The main difference is that on a sphere, the shortest path between two points is not a straight line, but a curve along the surface of the sphere. This means that when a vector is parallel transported along a path on a sphere, its direction will change in order to maintain its orientation relative to the surface.
The angle excess of 180° in parallel transport around a triangle on a sphere is a measure of the curvature of the surface. If the triangle is transported along a closed path and the angle excess is greater than 180°, it indicates that the surface is curved in that region. If the angle excess is exactly 180°, then the surface is considered to be flat in that region.
Geodesics are the shortest paths between two points on a curved surface. In parallel transport around a triangle on a sphere, the path that the vector takes to maintain its direction is a geodesic. This means that the angle excess of 180° is related to the curvature of the surface along the path of the triangle.
Parallel transport around a triangle on a sphere has applications in various fields such as mathematics, physics, and computer graphics. It is used to study properties of curved surfaces and can also be applied in navigation and mapping, as well as in computer graphics to create realistic 3D models of curved objects.