# Conifolds and Exotic Spheres

First of all, a disclaimer: I'm a physicist trying to understand mathematical aspects of some solutions I've encountered while studying string theory, and am certainly not a mathematician of any sort. In particular I have never taken a class on differential geometry or topology, hence my following question.

I'm reading an older paper in mathematical physics, and I'm trying to understand it. The paper can be found here: http://www.sciencedirect.com/science/article/pii/055032139090577Z

The authors are constructing 5 dimensional Einstein spaces, which they call $\mathcal{N}_{pq}$. I've also seen them denoted $T^{pq}$. I apologize that I don't have a name for them. The spaces are

$ds^2_{pq} = \lambda^2 \left(d\psi+p\cos\theta_1 d\phi_1 + q \cos\theta_2 d\phi_2 \right)^2 + \Lambda_1^{-1} \left(d\theta_1^2 + \sin^2\theta_1 d\phi_1^2 \right) + \Lambda_{2}^{-1} \left(d\theta_2^2 + \sin^2\theta_2 d\phi_2^2 \right).$

Here the lambda constants are fixed to make the space Einstein, $R_{ab} = 4 g_{ab}$:

$4 = \frac{1}{2}\lambda^2 \left[ (p\Lambda_1)^2 + (q \Lambda_2)^2 \right] = \Lambda_1 - \frac{1}{2} \left(\lambda p \Lambda_1\right)^2 = \Lambda_2 - \frac{1}{2} \left(\lambda q \Lambda_2\right)^2.$

For two particular choices of $(p,q)$ these spaces are furthermore fibre bundles over $S^2 \times S^3$. These are $(p,q) = (1,1)$ and $(1,0)$.

I'm confused about the relationship between these two spaces. The authors show that the two spaces are indeed distinct geometries because they have different volumes ($\text{vol}_{11}/\text{vol}_{10} = 2^{13/2} 3^{-5}$). The authors also say that these two spaces are diffeomorphic but that no coordinate transformation exists which brings one space into the other (and they use the unequal volumes to support this conclusion).

I'm confused by this. What does it mean for two spaces to be diffeomorphic and yet "distinct"? I recently learned about exotic spheres (https://en.wikipedia.org/wiki/Exotic_sphere). These are spheres that are homeomorphic but not diffeomorphic to the standard sphere (defined by the embedding $\sum_{i=1}^n x_i^2 = 1$ in $\mathbb{R}^n$). I'm wondering if perhaps the authors used out-dated terminology and the modern and more precise statement would be that $\mathcal{N}_{11}$ and $\mathcal{N}_{10}$ are homeomorphic but not diffeomorphic (just like the connection between exotic spheres and the standard sphere). Is this correct?

• I haven't read carefully, but they do not say that the manifolds are not diffeomorphic. They say that the geometry (which I interpret to mean the Riemannian geometry) is distinct - that is, that the Riemannian manifolds are not isometric. An isometry of Riemannian manifolds preserves the volume of the manifold; a diffeomorphism does not necessarily. One can put many non-isometric Riemannian metrics on the same smooth manifold; for instance, one has the round metric on the sphere, which has constant curvature, and one can scale the metric by some positive smooth function to get somethign new.
– user98602
Apr 10, 2015 at 19:57
• You're right, in fact they do explicitly say that the spaces are diffeomorphic. I was wondering if perhaps they were using the term incorrectly. Apr 10, 2015 at 20:03
• @MikeMiller: How is volume defined? Can you please suggest some literature for manifold volume constancy? Apr 10, 2015 at 20:52
• The poor man's (physicists') way is this: if you have the metric, and the range of coordinates, you could simply do the integral $\int \sqrt{g} d^d x$ yourself. Apr 10, 2015 at 21:05
• @Narasimham: The volume of a compact manifold is the integral of the Riemannian volume form. An isometry pulls back the volume form of the second manifold to the volume form of the first, so the two Riemannian manifolds have the same volume.
– user98602
Apr 10, 2015 at 21:05

Your two manifolds are Riemannian manifolds: smooth manifolds equipped with a Riemannian metric. The volume of a Riemannian manifold is preserved under isometry: it's defined by integrating the volume form over the manifold, and an isometry $M \to N$ pulls back the volume form of $N$ to the volume form of $M$. But this is far from true just for diffeomorphisms.
For instance, the $2$-sphere of any radius is diffeomorphic to the 2-sphere of any other radius; but the volume of $S^2(r)$ is $4\pi r^2$. So we can give a sphere a Riemannian metric of any given volume.