- #1

fab13

- 318

- 6

1*)

Why does one say that parallel transport preserves the value of dot product (scalar product) between the transported vector and the tangent vector ?

Is it due to the fact that angle between the tangent vector and transported vector is always the same during the operation of transport (which is the definition of parallel transport) ?

2*) From the following link http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/Geodesics.pdf and more especially the following extract ;

I don't understand the first relation, i.e : $$(\mathbf{t} \cdot \mathbf{\nabla})\,\mathbf{v}=0\quad\quad(1)$$

which actually is equal to : $$t^{i}\partial_{i}v^{j}=0\quad\quad(2)$$

How can we demonstrate the equation (1) above ?

Is there a link with my first question, i.e it relates to the conservation of dot product value between $$\mathbf{t}$$ tangent vector and $$\mathbf{v}$$ vector ?

Thanks for your help