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I Notations used with vector field and dot product

  1. Jan 22, 2017 #1
    Hello,

    I try to understand the following demonstration of an author (to proove that dot product is conserved with parallel transport) :
    ------------------------------------------------------------------------------------------------------------------------
    Demonstration :


    By definition, the parallel transport of ##e \in T{p}M## along a path ##\gamma(t), \gamma(0) = p## is the unique vector fields ##X_t## with ##X_t \in T_{\gamma(t)}M## such that ##\nabla_{\overset{\cdot}{\gamma}}X = 0 ## and ##X_0 = e##.

    Now, by definiton your connection is compatible with the metric, i.e ##Z \langle X,Y \rangle = \langle \nabla_Z X, Y\rangle + \langle X, \nabla_Z Y\rangle## for any vector field ##Z##.

    Thus taking ##Z = d\gamma/dt##, we obtain that ##\frac{d}{dt}\langle X,Y \rangle = \langle \nabla_{\overset{\cdot}{\gamma}} X, Y\rangle + \langle X, \nabla_{\overset{\cdot}{\gamma}} Y\rangle = 0## since ##X,Y## are parallel vector. Thus ##\langle X, Y \rangle = \langle X_0, Y_0 \rangle## as wished.
    ------------------------------------------------------------------------------------------------------------------------

    Unfortunately, I am not an expert in tensor calculus but I know some basics like the definition of covariant derivative of a vector ##V## along a geodesic - like with this notation :

    $$\nabla_{i}V^{j}=\partial_{i}V^{j}+V^{k}\Gamma_{ik}^{j}\quad\quad(1)$$

    and the absolute derivative : $$D\,V^{j}=(\nabla_{i}V^{j})dx^{i}\quad\quad(2)$$

    Could give me the link between this equation (##Z \langle X,Y \rangle = \langle \nabla_Z X, Y\rangle + \langle X, \nabla_Z Y\rangle## ) and the equation (1) or (2).

    Moreover, author defines ##Z## like ##\text{d}\gamma/\text{d}t## but after, he only takes ##\text{d}/\text{d}t## in :

    $$\frac{d}{dt}\langle X,Y \rangle = \langle \nabla_{\overset{\cdot}{\gamma}} X, Y\rangle + \langle X, \nabla_{\overset{\cdot}{\gamma}} Y\rangle = 0$$

    Author says that ##Z## is a vector field : is it an operator or a vector field ?

    And what about ##\langle X,Y\rangle## ? is it the dot product of ##X## and ##Y## ?

    Can one write :

    $$\langle X,Y\rangle=g_{ij}X^{i}Y^{j}$$

    with ##g_{ij}## the metrics ???

    Thanks for your help
     
  2. jcsd
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