Solving 2D Riemannian Metric Killing Vector Eqns.

In summary, to prove that a vector is a Killing vector, you need to calculate the connection coefficients for the specified metric and then use them to calculate the Lie derivative of the metric tensor with respect to the vector. If the result is equal to zero, then the vector is a Killing vector. You can evaluate each term of the tensor explicitly for the variables given.
  • #1
jiggers
1
0
noob here
* indicates multiply (or 'operate on'), d_c is partial derivative w.r.t. c

tensor indices have always troubled me, my problem this time is I am trying to prove a vector E = (-y*d_x +x*d_y) is a killing vector after having computed the connection coefficients for 2-d riemannian manifold, diagonal metric from ds^2 = f(x,y)*(dx^2 + dy^2)

Im looking at the Lie derivative statement of Killing's eqns,
E^c*d_c*g_ab - g_ac*d_c*E^b - g^cb*d_c*E^a

how do i do that explicitly, showing each term for x and y? how can there be terms involving a,b and c, when i only have x and y to work with?
any links to examples of "show this is a killing vector" would be a great help.
 
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  • #2
To answer your question, first you need to calculate the connection coefficients for the metric you have specified. This can be done by writing the metric in the form $g_{ij}=e^{\lambda(x,y)}(\partial_i x \partial_j x + \partial_i y \partial_j y)$ and then computing the Christoffel symbols from this. Once you have the connection coefficients, you can use them to calculate the Lie derivative of the metric tensor. The Killing equation states that if the Lie derivative of the metric vanishes, then the vector is a Killing vector. So, for your example, you would calculate the Lie derivative of the metric tensor $g_{ab}$ with respect to the vector $E^c = (-y\partial_x + x\partial_y)$. The result will be a tensor of the form $\mathcal{L}_E g_{ab}=E^c \partial_c g_{ab}-g_{ac}\partial_b E^c-g_{bc}\partial_a E^c$. You can then evaluate each term of the tensor explicitly for $x$ and $y$. For example, the first term will be $E^c \partial_c g_{ab}= (-y\partial_x + x\partial_y)\partial_c (e^{\lambda(x,y)}(\partial_i x \partial_j x + \partial_i y \partial_j y))$. Once you have evaluated all the terms, you can check to see if the result is equal to zero. If it is, then the vector is a Killing vector. If not, then the vector is not a Killing vector. I hope this helps!
 

What is a 2D Riemannian metric?

A 2D Riemannian metric is a mathematical concept used in the study of geometry and differential equations. It is a way to measure distances and angles in a two-dimensional space, such as a flat plane or a curved surface.

What are Killing vector equations?

Killing vector equations are a set of equations used to determine the symmetries of a Riemannian metric. They are named after the mathematician Wilhelm Killing and are used to find vector fields that preserve the metric's properties, such as distances and angles.

Why is solving 2D Riemannian metric Killing vector equations important?

Solving these equations allows us to find the symmetries of a Riemannian metric, which can provide valuable information about the geometry of a space. This can be useful in various fields such as physics, where symmetries play a crucial role in understanding the behavior of physical systems.

What are some applications of solving 2D Riemannian metric Killing vector equations?

Some applications of solving these equations include studying the geometry of curved surfaces in mathematics, finding exact solutions to certain differential equations in physics, and understanding the symmetries of spacetime in general relativity.

Are there any challenges in solving 2D Riemannian metric Killing vector equations?

Yes, there can be challenges in solving these equations, as they often involve complex mathematical techniques and can be time-consuming. Additionally, the solutions may not always be unique, and it may be difficult to determine which solution is the most relevant for a given problem.

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