Litterature on small or compact dimensions?

[center o bass]

Litterature on "small" or "compact" dimensions?

Hi! I'm reading some Kaluza-Klein theory which is an extension of normal 4D GR to a 5D spacetime in which the fifth dimension is a "small" or "compact" extra spatial dimension. I've found loads of literature on the differential geometry of hypersurfaces in higher-dimensional spaces, but I've not found anything on (the diff. geometry of) compact or small spaces. Does anyone have any recommendations on what to read in this regard?

Thanks!

 

[fzero]

Compact refers to a specific mathematical property that roughly means that the manifold has a finite volume. "Small" is more of a physical term that compares the size of the compact manifold to other length scales involved in the physical situation. So a compact manifold will be "small", if the length scale associated with its volume is small compared to the inverse energy (in appropriate units) of a typical process that we are studying.

From the physical perspective, Nakahara, Geometry, Topology and Physics, is reasonably well-written and covers almost all of the topics that you would need to know. At least it would make a good foundation for more detailed study from more formal or advanced texts. From a mathematics perspective, I'm not as familiar with lower-level texts. Bott and Tu, Differential Forms in Algebraic Topology, focuses on algebraic topology concepts of homology and homotopy, but doesn't include any Riemannian geometry. Very basic geometry concepts are covered in Spivak's Calculus on Manifolds, with more advanced topics appearing in vol 1 of Spivak's Differential Geometry and Do Carmo's Riemannian Geometry. There are a couple of books by Milnor ("Topology from the differentiable viewpoint" and "Morse theory") that are well-recommended but I am unfortunately not too familiar with.

 

[center o bass]

Compact refers to a specific mathematical property that roughly means that the manifold has a finite volume. "Small" is more of a physical term that compares the size of the compact manifold to other length scales involved in the physical situation. So a compact manifold will be "small", if the length scale associated with its volume is small compared to the inverse energy (in appropriate units) of a typical process that we are studying.

From the physical perspective, Nakahara, Geometry, Topology and Physics, is reasonably well-written and covers almost all of the topics that you would need to know. At least it would make a good foundation for more detailed study from more formal or advanced texts. From a mathematics perspective, I'm not as familiar with lower-level texts. Bott and Tu, Differential Forms in Algebraic Topology, focuses on algebraic topology concepts of homology and homotopy, but doesn't include any Riemannian geometry. Very basic geometry concepts are covered in Spivak's Calculus on Manifolds, with more advanced topics appearing in vol 1 of Spivak's Differential Geometry and Do Carmo's Riemannian Geometry. There are a couple of books by Milnor ("Topology from the differentiable viewpoint" and "Morse theory") that are well-recommended but I am unfortunately not too familiar with.

Thanks for that clarification. I have read some of Nakahara's book, but it does not seem to contain much about hypersurfaces, connections on hypersurfaces etc. To be a little bit more specific I'm wondering about the relations between a connection on the compact dimension in relation to the connection on the 4D spacetime as well as the relation between the curvatures; just as one obtain relations between the Riemann-tensor in the full manifold and the Riemann-tensor on the hypersurface through Gauss' theorema egregium.

 

[WannabeNewton]

Do you know what a compactification is mathematically to start with? See e.g. chapter 5 of Munkres' topology text for an introduction. It's akin to completion of metric spaces in a loose sense. The notion of compactness itself is best thought of as a tool to turn local topological properties into global properties (in fact you can prove this); the notion of "finite volume" works fine in a loose sense when you're in $\mathbb{R}^{n}$ and Heine-Borel holds but for arbitrary topological spaces and even manifolds, the concept is not nearly as trivial.

Anyways, I would suggest you check out Frankel "The Geometry of Physics".

 

[fzero]

Are you sure that the papers you've turned up are explicitly talking about embedding noncompact submanifolds as opposed to compact ones? It seems to me, just based on experience, that it's far more likely to find a discussion of compact manifolds, for a number of reasons.

In any case, there is a decent review of KK theories, by Bailin and Love that might be useful for you. Perhaps there are some references there to more general geometry papers.