I'm in a logical loop here: 1. A tensor undergoes parallel transport if, as it moves through a manifold, its covariant derivative is zero. 2. Covariant derivative describes how a tensor changes as it moves through a manifold. 3. A tensor undergoes change as it moves if it does not parallel transport. So how do I get out of this loop? I have an intuitive sense of parallel transport, but I do not know how to describe it mathematically except by using the definition above (1). Perhaps the answer lies in the calculus of variations?