- #1
strauser
- 37
- 4
I'm trying to understand connections and their arbitrariness.
Many diff. geom. books or webpages appear to be contradictory. A chapter or page on connections may start off stressing that a connection is an arbitrary method of mapping between tangent spaces, then shortly after, show that nice picture of the "parallel transport" of an arrow on the surface of the sphere.
Now, this confuses me a little. If a connection is genuinely arbitrary, shouldn't we see various pictures of "parallel transport", where the arrow appears to rotate or stretch (smoothly, I guess) as it moves from tangent space to tangent space? Yet I've never seen this - I've only seen the somewhat cliched image where the arrow is transported such that it is parallel in the eyes of an observer sitting outside the manifold (from the pole to the equator, along the equator, and back to the pole)
Am I right in assuming that this picture in fact is showing not some arbitrary connection but specifically a Levi-Civita connection? i.e. that it tacitly assumes some metric on the sphere from which the connection coefficients have been calculated? If so, doesn't this image give a very confusing and limited intuition for what a connection is allowed to do? Or am I myself confused?
Many diff. geom. books or webpages appear to be contradictory. A chapter or page on connections may start off stressing that a connection is an arbitrary method of mapping between tangent spaces, then shortly after, show that nice picture of the "parallel transport" of an arrow on the surface of the sphere.
Now, this confuses me a little. If a connection is genuinely arbitrary, shouldn't we see various pictures of "parallel transport", where the arrow appears to rotate or stretch (smoothly, I guess) as it moves from tangent space to tangent space? Yet I've never seen this - I've only seen the somewhat cliched image where the arrow is transported such that it is parallel in the eyes of an observer sitting outside the manifold (from the pole to the equator, along the equator, and back to the pole)
Am I right in assuming that this picture in fact is showing not some arbitrary connection but specifically a Levi-Civita connection? i.e. that it tacitly assumes some metric on the sphere from which the connection coefficients have been calculated? If so, doesn't this image give a very confusing and limited intuition for what a connection is allowed to do? Or am I myself confused?