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Sorry if this is a dumb question, but what

*exactly*do we mean by the term parallel transport? Is it just the physicist's way of saying isometry?

Also, in my class we have just defined geodesics, and we're told that having a geodesic curve cis equivalent to demanding that the unit tangent vector be parallel transported along c. This confuses me. I can read the proof of this, and it's fine, but I don't feel any physical intuition developing here. Can someone explain, as physically as possible, why these two conditions are equivalent?

Thanks.