- #1
fab13
- 318
- 6
Hello, I have 2 questions regarding similar issues :
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
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