Orthonormal Frame: is $\nabla_{e_1}e_j$ 0 if i≠j?

[forumfann]

Could anyone help me on this question:
Suppose ${e_i}$ is an orthonormal frame, is $\nabla_{e_1}e_j$ is 0 if i is not equal to j?

Any answers or suggestion will be highly appreciated.

 

[HallsofIvy]

Could anyone help me on this question:
Suppose ${e_i}$ is an orthonormal frame, is $\nabla_{e_1}e_j$ is 0 if i is not equal to j?

Any answers or suggestion will be highly appreciated.

First, use [ itex ] and [ /itex ] or [ tex ] and [ itex ] (without the spaces) on this forum rather that "$". So your question is
"Suppose $\{e_i\}$ is an orthonormal frame, is $\nabla_{e_i}e_j$ is 0 if i is not equal to j?"

Okay, what does $\nabla_{ei}e_j$ mean?

 

[forumfann]

$\nabla_{e_{i}}e_{j}$ is the Levi-Civita connection between $e_{i}$ and $e_{j}$. Does $\nabla_{e_{i}}e_{j}$ equal 0 if i is not equal to j?

 

[Ben Niehoff]

No, not in general, because the frame can rotate.