Notations used with vector field and dot product

[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 ?

Thanks for your help

 

[jvicens]

Thank you for your question. I will do my best to explain the link between the equation $Z \langle X,Y \rangle = \langle \nabla_Z X, Y\rangle + \langle X, \nabla_Z Y\rangle$ and equations (1) and (2) that you mentioned.

Firstly, the equation $Z \langle X,Y \rangle = \langle \nabla_Z X, Y\rangle + \langle X, \nabla_Z Y\rangle$ is known as the Leibniz rule for covariant derivatives. It is a fundamental property of the covariant derivative, which is the generalization of the usual derivative to curved spaces. This rule essentially tells us how to differentiate a tensor product of two vector fields with respect to another vector field.

Now, let's look at equation (1) and (2). Equation (1) is the definition of the covariant derivative, where $\Gamma_{ik}^{j}$ are the Christoffel symbols, which represent the connection coefficients of the metric. These coefficients tell us how the basis vectors of a tangent space change as we move along a curve. Equation (2) is the absolute derivative, which is simply the covariant derivative written in terms of differential forms.

To understand the link between these equations, we need to understand the role of the connection in the covariant derivative. The connection is a way of comparing vectors at different points in a curved space. It tells us how to move vectors from one tangent space to another along a curve. In the case of parallel transport, the connection ensures that the vector remains parallel to itself as it is transported along the curve.

Now, let's look at the author's demonstration. The vector field $Z$ is defined as $\text{d}\gamma/\text{d}t$, which is the tangent vector to the curve $\gamma(t)$ at each point. This vector field is a way of parametrizing the curve. The notation $\text{d}/\text{d}t$ is simply the operator for taking the derivative with respect to the parameter t.

Finally, $\langle X,Y \rangle$ is indeed the dot product of the vector fields $X$ and $Y$. In this context, $\langle X,Y\rangle=g_{ij}X^{i}Y^{j}$ is a shorthand notation