# Compactability and Killing vectors

## Main Question or Discussion Point

I have read statements like "assume that there exists a killingvector $\xi$ that makes it possible to compactify the space in it's direction." It's not hard to find examples of compact "directions" with a corresponding killing vector (rotational direction on a sphere), but there are also examples of killing vectors pointing in a non-compact direction (translations in $\mathbb{R}^n)$. Is there however a requirement that there exists a killing vector in the given direction in order for it to be "compactable"? I.e. does the non-existence of a killing-vector imply that the "direction" is non-compactable? If so why?

## Answers and Replies

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fzero
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There are certainly compact manifolds with no isometries at all. The most obvious examples are compact Riemann surfaces with genus $\geq 2$. Another are the Riemannian manifolds with special holonomy.

It might help to point out a specific reference to be sure, but I suspect that the reason for requiring the existence of a Killing vector is a crucial part of Kaluza-Klein theory, where the Killing vectors generate gauge symmetries in the effective theory on the noncompact directions. This generally leads to the study of homogeneous spaces, which have well-defined isometry and have lots of examples where explicit metrics can be written.

There are certainly compact manifolds with no isometries at all. The most obvious examples are compact Riemann surfaces with genus $\geq 2$. Another are the Riemannian manifolds with special holonomy.

It might help to point out a specific reference to be sure, but I suspect that the reason for requiring the existence of a Killing vector is a crucial part of Kaluza-Klein theory, where the Killing vectors generate gauge symmetries in the effective theory on the noncompact directions. This generally leads to the study of homogeneous spaces, which have well-defined isometry and have lots of examples where explicit metrics can be written.
Alright, so it is not necessary to have isometries in order to compactify a manifold, but is it sufficient?

WannabeNewton
Science Advisor
What definition of compactification is being used here? The simplest kind of compactification, which is the Alexandroff one-point compactification, only requires a locally compact Hausdorff space; all manifolds are locally compact and Hausdorff so that's always satisfied. More generally one has the following: http://en.wikipedia.org/wiki/Stone-Czech_compactification

There are certainly compact manifolds with no isometries at all. The most obvious examples are compact Riemann surfaces with genus $\geq 2$. Another are the Riemannian manifolds with special holonomy.

It might help to point out a specific reference to be sure, but I suspect that the reason for requiring the existence of a Killing vector is a crucial part of Kaluza-Klein theory, where the Killing vectors generate gauge symmetries in the effective theory on the noncompact directions. This generally leads to the study of homogeneous spaces, which have well-defined isometry and have lots of examples where explicit metrics can be written.
Fzero. Do you have knowledge about the subject of KK-theory? If so and if it's okay with you, I would very much like to ask you some questions about it :) I could not send you a private message though.

George Jones
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I have read statements like "assume that there exists a killingvector $\xi$ that makes it possible to compactify the space in it's direction."
Can you give an example?

fzero
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Fzero. Do you have knowledge about the subject of KK-theory? If so and if it's okay with you, I would very much like to ask you some questions about it :) I could not send you a private message though.
I have PMs turned off because I'd prefer keeping discussions public, for a number of reasons. I prefer that the information be available to others that might be searching for it and I prefer that other knowledgeable people have an opportunity to add to or correct the information.

I have a certain amount of knowledge of KK theory, as it has been tangentially related to other topics that I've researched, so I can try to answer additional questions. I would suggest posting a new thread for questions that are very different than the present one. While you can't PM, I am subscribed to any thread that I've posted in, so you could post a link to the thread here for example if I don't notice the new post. I will attempt to answer the question or let you know that I cannot.

George Jones
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Can you give an example?
Okay, I found an example. From page 481 of the Chapter "Kaluza-Klein Theory" in the book "Einstein's General Theory of Relativity with Modern Applications in Cosmology" by Gron and Hervik:" Assume also there is one spatial Killing vector. This makes it possible to compactify the space in that direction, and make it as small as needed."

Fortunately, Gron and Hervik expands on some of the points fzero, and I encourage you (center o bass) to read sections 15.1 Lie groups and Lie algebras, 15.2 Homogeneous spaces, and 15.6 Constructing compact quotients.

The metric is local, and we are somewhat free to specify the global topology by taking topological quotients.

Okay, I found an example. From page 481 of the Chapter "Kaluza-Klein Theory" in the book "Einstein's General Theory of Relativity with Modern Applications in Cosmology" by Gron and Hervik:" Assume also there is one spatial Killing vector. This makes it possible to compactify the space in that direction, and make it as small as needed."

Fortunately, Gron and Hervik expands on some of the points fzero, and I encourage you (center o bass) to read sections 15.1 Lie groups and Lie algebras, 15.2 Homogeneous spaces, and 15.6 Constructing compact quotients.

The metric is local, and we are somewhat free to specify the global topology by taking topological quotients.
Thanks for the tip. I have briefly read some of the arguments, but I will dive deeper into them when I have time. As you also mentioned I understood that the killing vector (i.e symmetry) was necessary in order to make point identification necessary for compactifying a non-compact dimension.

I've read arguments that suggest that compactification of dimensions might have occured in the early universe 'spontaneously'.
But then I wonder; does the argument from Gron and Hervik's book imply that a dimension has already to be symmetric before 'spontaneous compactification' can occur.

I have PMs turned off because I'd prefer keeping discussions public, for a number of reasons. I prefer that the information be available to others that might be searching for it and I prefer that other knowledgeable people have an opportunity to add to or correct the information.

I have a certain amount of knowledge of KK theory, as it has been tangentially related to other topics that I've researched, so I can try to answer additional questions. I would suggest posting a new thread for questions that are very different than the present one. While you can't PM, I am subscribed to any thread that I've posted in, so you could post a link to the thread here for example if I don't notice the new post. I will attempt to answer the question or let you know that I cannot.
Thank you! I really appreciate it. As you suggested I have posted a new thread; this time on the effects of compactification and if it somehow prevents one from 'gauging' away the maxwell tensor. It can be found here :

https://www.physicsforums.com/showthread.php?t=728572