Does parallel transport preserve length?
Essentially, when a metric is present on a manifold, the connection must be metric-compatible, which means not just length preservation, but also angle preservation between two vectors as are they are parallel transported simultaneously along the same path.
What is parallel displacement?
From Encyclopedia of Mathematics. A special case of a motion in which all points of the space are transferred in one and the same direction (along a line in that space) over one and the same distance.
Does parallel transport preserve angles?
How to show how a vector is parallel transported around a sphere?
Choose the radius of the sphere so that it sits in the mouth of the cone as shown in figure (a), so that the curved edge of the shape from figure (b) touches the sphere along a small circle. Then the red arrows will be tangent the the sphere along that small circle, and they will show how a vector is parallel transported around that small circle.
What is parallel translation of a vector on a surface?
Parallel translation of a vector on a surface: parallels of the sphere Related 9 how does parallel transport work on the sphere 6 Parallel transport along a 2-sphere. 1 Parallel transport for infinitesimal displacement
How do you rotate the cone of a parallel transport curve?
And for the cone, there’s an easy way of seeing that there is indeed a rotation of the transported vector with respect to the tangent vector of the curve: just cut the cone open and lay it flat on the table, so that parallel transport becomes simply ordinary parallel transport on the plane.
What happens when you travel in a straight line on a sphere?
This is the same effect you get on a flat plane when you travel in a straight line across a field of parallel vectors. An important difference between a flat plane and the surface of a sphere, however, is that there is no uniform parallel field of vectors on the surface of the sphere.