Re: Connection not a tensor..

mike.james wrote:
> I've been thinking about what it means that the connection coefficients
> don't transform like a tensor. Given that we start off with the idea
> that thing that have a co-ordinate independent reality are tensors i.e.
> transform like tensors does the fact that the connection isn't a tensor
> mean that it isn't a "physical thing" or does it mean that other
> transformation laws define physical things.
> Given that a connection does define things that are independent of
> co-ordinates such as curvature, parallel transport etc. presumably the
> connection is as real as a tensor and hence the transformation law that
> it obeys is "as good as" the tensor transformation law.
> Or is there a bigger picture in which the connection is part of a tensor?

You can think of the connection as providing a kind of correction
term: you add this to a coordinate derivative (which also isn't
a tensor) and the result is then a tensor---i.e. a nice geometric
notion of derivative.

.

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