# I Notations used with vector field and dot product

1. Jan 22, 2017

### fab13

Hello,

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 ???