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Math Help - Question about linear connection and covariant derivative along curves

  1. #1
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    Question about linear connection and covariant derivative along curves

    Hello,

    Let M be a Riemannian Manifold, \nabla Levi-Civita-Connection and \frac{D}{dt} the induced covariant derivative along curves.
    Let c:J\rightarrow M be a curve and L\subset J an intervall, Z a vector field along c.

    Can someone help me find out why the following statement is true?


    (\frac{D}{dt}Z)|_L=\frac{D}{dt}(Z|_L) (The left \frac{D}{dt} is the covariant derivative along c and the right \frac{D}{dt} is the covariant derivative along c|_L).

    I already know that if X and Y are vector fields along c and if X|_{(t_0-\varepsilon,t_0+\varepsilon)}=Y|_{(t_0-\varepsilon,t_0+\varepsilon)} then (\frac{D}{dt}X)(t_0)=(\frac{D}{dt}Y)(t_0).

    Regards,
    engmaths
    Last edited by engmaths; December 8th 2012 at 04:17 AM.
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  2. #2
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    Re: Question about linear connection and covariant derivative along curves

    Suppose L is open so that for any t0 in L, there is a small r such that (t0-r, t0+r) is in L.
    If L is not open, for example L=[a,b), we can only define D/dt(Z|L) from the right side of a, thus there is a small r such that [a, a+r) is in L so the statement still holds.
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  3. #3
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    Re: Question about linear connection and covariant derivative along curves

    Hello xxp9,

    I'm not quite sure if you understood my problem.

    If we assume that L is open, why is the statement true then?

    Somehow I don't see a way to compare the left \frac{D}{dt} (the covariant derivative along c) and the right \frac{D}{dt} (the covariant derivative along c|_L).
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  4. #4
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    Re: Question about linear connection and covariant derivative along curves

    As you said if two vector fields are identical in an interval around a point along the curve,, they covariant derivative are identical at that point.
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  5. #5
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    Re: Question about linear connection and covariant derivative along curves

    I got another question concerning this topic:

    Assume you have a smooth curve c:J->M that admits an extension to a smooth curve definied on a larger intervall.
    Now, is then every vector field along c extendible to a vector field along the extension of c?
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  6. #6
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    Re: Question about linear connection and covariant derivative along curves

    since we can always express a vector field in terms of its components, which are smooth functions, based on the coordinates vectors, the question is equivalent to:
    Given a smooth function defined on [a, b], can we extend it to a larger interval?
    The answer is yes.
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