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

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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.

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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

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# I Notations used with vector field and dot product

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