§ Connections, take 2
- I asked a math.se question about position, velocity, acceleration that recieved a great answer by
peek-a-boo. Let me try and provide an exposition of his answer.
- Imagein a base manifold , say a circle.
- Now imagine a vector bundle over this, say 2D spaces lying above each point on the circle. Call this
- What is a connection? Roughly speaking, it seems to be a device to convert elements of into elements of .
- We imagine the base manifold (circle) as horizontal, and the bundle as vertical. We imagine as vectors lying horizontal on the circle, and we imagine as vectors lying horizontal above the bundle. So something like:
- So the connection has type . Consider a point in the base manifold.
- Now think of the fiber over .
- Now think of any point in the fiber of .
- This gives us a map , which tells us to imagine a particle following its brother in . If we know the velocity , we can find the velocity of the sibling upstrairs with .
- In some sense, this is really like path lifting, except we're performing "velocity lifting". Given a point in the base manifold and a point somewhere upstairs in the cover (fiber), we are told how to "develop" the path upstairs given information about how to "develop" the path downstairs.
- I use "develop" to mean "knowing derivatives".
§ Differentiating vector fields along a curve
- Given all of this, suppose we have a curve and a vector field over the curve such that the vector field lies correctly over the curve; . We want to differentiate , such that we get another .
- That's the crucial bit, and have the same type, and this is achieved through the connection. So a vector field and its derivative are both vector fields over the curve.
- How do we do this? We have the tangent mapping .
- We kill off the component given by pushing forward the tangent vector at the bundle location via the connection. This kills of the effect of the curving of the curve when measuring the change in the vector field .
- We build .
- We now have a map from to , but we want a map to . What do?
- Well, we can check that the vector field we have created is a vertical vector field, which means that it lies entirely within the fiber. Said differently, we check that it pushes forward to the zero vector under projection, so will be zero for the image of .
- This means that lies entirely "inside" each fiber, or it lies entirely in the tangent to the vector space (ie, it lives in ), instead of living in the full tangent bundle where it has access to the horizontal components.
- But for a vector space, the tangent space is canonically isomorphic to the vector space itself! (parallelogram law/can move vectors around/...). Thus, we can bring down the image of from down to !
- This means we now have a map .
- But we want a . See that the place where we needed a was to produce