Notations used with vector field and dot product

In summary: The metric tensor tells us how to measure distances and angles in the curved space. So, in summary, the author is using the Leibniz rule to show that the dot product is conserved under parallel transport because of the compatibility of the connection with the metric.I hope this helps to clarify the concepts for you. Please let me know if you have any further questions. In summary, the author uses the Leibniz rule to show that the dot product is conserved under parallel transport because of the compatibility of the connection with the metric. The connection is a way of comparing vectors at different points in a curved space, and the metric tensor tells us how to
  • #1
fab13
318
6
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
 
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  • #2

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
 

Related to Notations used with vector field and dot product

What is a vector field?

A vector field is a mathematical concept used in physics and engineering to describe the behavior of vector quantities in space. It is a function that assigns a vector to every point in a given space, such as a two-dimensional plane or three-dimensional space.

What is a dot product?

A dot product is a mathematical operation performed on two vectors to produce a scalar quantity. It is also known as the inner product or scalar product. It involves multiplying the magnitudes of the two vectors and the cosine of the angle between them.

What are some notations used with vector fields?

Common notations used with vector fields include arrow notation, where an arrow is placed above the variable representing the vector, and boldface notation, where the vector is represented by a boldface letter. Other notations include component notation, where the components of the vector are written in brackets, and index notation, where the vector is represented by a subscripted letter.

How is the dot product represented in different notations?

The dot product can be represented in different notations depending on the chosen coordinate system. In component notation, it is written as a sum of the products of the corresponding components of the two vectors. In index notation, it is written as the product of the components with the same index. In geometric notation, it is represented by the product of the magnitudes of the vectors and the cosine of the angle between them.

What is the purpose of using notations with vector fields and dot product?

Notations are used to represent vector fields and dot product in a concise and consistent manner. They also allow for easy manipulation and calculation of vectors and their properties. Different notations may be used for different purposes, such as theoretical analysis or practical applications in specific fields of study.

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