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Orthonormal frame

  1. Sep 5, 2009 #1
    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.
     
  2. jcsd
  3. Sep 5, 2009 #2

    HallsofIvy

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

    Okay, what does [itex]\nabla_{ei}e_j[/itex] mean?
     
  4. Sep 5, 2009 #3
    [itex]\nabla_{e_{i}}e_{j}[/itex] is the Levi-Civita connection between [itex]e_{i}[/itex] and [itex]e_{j}[/itex]. Does [itex]\nabla_{e_{i}}e_{j}[/itex] equal 0 if i is not equal to j?
     
  5. Sep 5, 2009 #4

    Ben Niehoff

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