story

I’ve been trying to make my way through some concepts in differential geometry and tensor calculus, and I’ve been having difficulty visualizing the effect of a changing metric tensor.

My intuition has been to picture manifolds embedded in an Euclidean space \(\mathbb{R}^n\), and this appears to be the approach taken in most books. However, it is not immediately clear that this intuitive picture is equivalent to the typical treatment of a manifold as a topological space described with coordinate charts. These two pictures are known as the extrinsic and intrinsic pictures respectively.

This post will focus on isometric embeddings, as they seem more tangible.

Here’s my attempt at visualizing this concept through embedding of a 2D manifold equipped with an arbitrary metric tensor in \(\mathbb{R}^3\). I also calculate a few other quantities on the manifold like the Levi-Civita connection and the Riemann tensor and the Ricci things to help get a better understanding of how that all relates to the metric.

hand-wavy introduction to manifolds

While reading this, please keep in mind that I’m probably the least qualified person to be writing about this subject.

I’m going to attempt to motivate and explain a few terms that will be key to limiting the scope of this post.

‘Riemannian’

A Riemannian manifold is a manifold equipped with a Riemannian metric. A Riemannian metric allows for the definition of an inner product on the tangent space \(T_pM\) on every point \(p\) on the manifold \(M\).

More specifically, the Riemannian metric is a positive definite map between two elements of the tangent space to the reals.

\[ g_p : T_pM \times T_pM \mapsto \mathbb{R}, \forall p \in M \]

If a basis is specified, \(g_p\) can be expressed as a positive definite \(m \times m\) matrix, where \(m\) is the dimension of the manifold.

‘embedding’

An embedding in topology between two topological spaces \(X\) and \(Y\) is a continuous injective mapping

\[ f : X \mapsto Y \]

Embedding a manifold in a some canonical space such as \(\mathbb{R}^n\) is typically used as a way to visualize the manifold. For example, imagining the 2-sphere in \(\mathbb{R}^3\) can be viewed as an embedding.

In many descriptions the 2-sphere is defined explicitly in reference to \(\mathbb{R}^3\) (i.e. as the set of points equidistant from a reference point (as measured in \(\mathbb{R}^3\))); however, this picture is limiting and misleading in topology. It is important to see a manifold like the 2-sphere as an object independent of some external embedding. This allows for much more general geometries.

‘isometric’

When a manifold is equipped with a metric, it is possible to measure distances between points on the manifold. Isometric embeddings are those that preserve these distances when using the standard metric in \(\mathbb{R}^n\),

This post will focus on isometric embeddings as I think they provide more physical intuition. Imagine manifolds made of a flexible but non-stretchable material like paper instead of rubber.

‘compact’

There is a rigorous definition of compactness which I have not spent the time to understand, but the notion of compactness is quite intuitive. If there is a finite distance between any two points on a manifold, it is compact (e.g. sphere is compact, plane is not).

A less obvious fact about compact manifolds is that they are completely specified by their orientability and genus.

2-sphere example

Let’s start with a concrete example.

For the purposes of this post, I will be looking at 2D manifolds embedded in \(\mathbb{R}^3\) in the interest of being able to visualize the manifold. I will also be using the Euclidean metric of signature \((0,2)\) for simplicity.

Let’s choose some coordinates for our manifold

\[ \xi^a = \hat{\xi^a}(x^1,x^2) = \hat{\xi^a}(x^\mu) \]

where \(a \in {1,2,3}\) and \(\mu \in {1,2}\). I’m using largely standard notation so I will gloss over the finer details.

metric tensor

The metric tensor \(g_{\mu\nu}\) allows us to measure arclengths. I usually remember it in the following context

\[ ds^2 = g_{\mu\nu} dx^\mu dx^\nu \]

Of course, the metric is also related to the covariant basis, which we can easily obtain now that we have a coordinate system

\[ g_{\mu\nu} = \vec{e_\mu} \cdot \vec{e_\nu} = \frac{\partial \xi^a}{\partial x^\mu} \frac{\partial \xi^b}{\partial x^\nu} \eta_{ab} \]

where \(\eta_{ab}=I_2\), the \(2 \times 2\) identity matrix in this case (since we’re using a Euclidean metric signature).

To make all of this more concrete, let’s look at the 2-sphere with the typical spherical coordinates

\[ \xi^1 = \rho \sin{x^1} \cos{x^2} \]

\[ \xi^2 = \rho \sin{x^1} \sin{x^2} \]

\[ \xi^3 = \rho \cos{x^1} \]

The metric tensor in these coordinates is

\[ (g)_{\mu\nu} = \begin{bmatrix} \rho^2 & 0 \\ 0 & \rho^2\sin^2{x^1} \\ \end{bmatrix} \]

Levi-Civita Connection

Let’s quickly compute the Levi-Civita connection for the 2-sphere. I will try to demonstrate the geometric intuition for this later.

\[ \Gamma^\sigma_{\mu\nu} = \frac{1}{2} g^{\sigma\rho}(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\rho\mu} - \partial_\rho g_{\mu\nu}) \]

where \(\partial_\mu \equiv \frac{\partial}{\partial x^\mu}\).

Applying the above equations,

\[ \Gamma^1_{22} = -\sin{x^1} \cos{x^1} \]

and

\[ \Gamma^2_{21} = \Gamma^2_{22} = \cot{x^1} \]

with all the other components of the connection equal to zero.

Riemann curvature tensor

Now that we have the Christoffel symbols, let’s also quickly compute the curvature tensor for the 2-sphere. The Riemann tensor can be derived from the Levi-Civita connection calculated above,

\[ R^\rho_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma} \]

Applying the above equations, the non-zero components of the Riemann tensor are

\[ R^1_{212} = -R^1_{221} = \sin^2{x^1} \]

and

\[ R^2_{112} = -R^2_{121} = -1 \]

So the Levi-Civita connection and the Riemann tensor are both specified for a manifold equipped with a metric. Now to help visualize the manifold, let’s try to embed it in \(\mathbb{R}^3\).

Ricci tensor

The Ricci tensor is closely related to the Riemann tensor,

\[ R_{\mu\nu} = R^\lambda_{\mu\lambda\nu} \]

For the 2-sphere,

\[ (R)_{\mu\nu} = \begin{bmatrix} 1 & 0 \\ 0 & \sin^2{x^1} \\ \end{bmatrix} \]

Ricci scalar

Since we also know the metric, we can get the Ricci scalar,

\[ R = g^{\mu\nu} R_{\mu\nu} \]

For the 2-sphere,

\[ R = \frac{2}{\rho^2} \]

going the other way

Is it possible in general to go from the intrinsic notion of a manifold equipped with a metric to an embedding in \(\mathbb{R}^3\)?

system of PDEs

Given a manifold equipped with a metric, \(g_{\mu\nu}\), we know that

\[ \frac{\partial \xi^a}{\partial x^\mu} \frac{\partial \xi^b}{\partial x^\nu} \eta_{ab} = g_{\mu\nu} \]

This is a system of non-linear partial differential equations. Embedding a \(m\)-dimensional manifold in a \(n\)-dimensional space, we naievely have \(m^2\) equations and \(n\) free variables. Of course, since we are dealing with a Riemannian metric, we have, in general, \(m(m+1)/2\) independent equations.

For the 2-sphere in \(\mathbb{R}^3\) and \((0,2)\) signagure described above, we have

\[ \left(\frac{\partial \xi^1}{\partial x^1}\right)^2 + \left(\frac{\partial \xi^2}{\partial x^1}\right)^2 + \left(\frac{\partial \xi^3}{\partial x^1}\right)^2 = \rho^2 \]

and

\[ \left(\frac{\partial \xi^1}{\partial x^2}\right)^2 + \left(\frac{\partial \xi^2}{\partial x^2}\right)^2 + \left(\frac{\partial \xi^3}{\partial x^2}\right)^2 = \rho^2\sin^2{x^1} \]

You can check that the standard embedding of the 2-sphere satisfies this system of PDEs. In this case we’re actually missing an equation because of the diagonal nature of \(\eta_{ab}\) and \(g_{\mu\nu}\).

Of course, in general, solving non-linear systems of PDEs is not easy. In fact even existence and uniqueness are difficult to ascertain in general.

the torus

Let’s quickly look the case of the flat torus.

A lot of confusion seems to stem from the overloading of seemingly familiar terms and ideas.

For example, in topology, the torus is the Cartesian product of two circles \(S^1 \times S^1\). This definition does not necessitate a metric. The torus can be equipped with an arbitrary metric later (or given one through embedding in \(\mathbb{R}^n\)). The typical geometric notion of a torus (the one I imagine) is the embedding in \(\mathbb{R}^3\), known as the ring torus. This embedding artificially imposes a metric on the torus, but care should be taken not to confuse this metric as property of the topological torus. For example, alternative embeddings of the torus are possible, such as the Clifford torus in \(\mathbb{R}^4\).

A metric can be assigned to the torus before embedding. For example, a torus can be given a flat metric that has zero curvature. This is known as a flat torus. It is understandably difficult to imagine what a flat torus would look like. By this, I mean it is uncertain whether or not we can find an isometric embedding of the flat torus in \(\mathbb{R}^3\) where we derive most of our intuition. Imagine bending a piece of paper, which has a flat metric, into a cylinder and then bringing the two ends of the cylinder together. Initially, this does not seem possible without stretching the paper. Is it possible to embed the flat torus in \(\mathbb{R}^3\)? Does smoothness have to be sacrificed? This is where Nash embedding theorems make an appearance.

Nash-Kuiper theorem

Roughly speaking, suppose we have a Riemannian manifold \((M,g)\), the Nash-Kuiper theorem says that we can find a \(C^1\) embedding, \(f: M^m \mapsto \mathbb{R}^n \) for \(n \ge m+1\).

This means that there is a continuously differentiable embedding of the flat torus (a 2-dimensional Riemannian manifold) into \(\mathbb{R}^3\). This is somewhat surprising.

Of course this theorem does not actually bring us any closer to finding a isometric embedding, but it is nice to know that one exists.

Recently, Borrelli et al. (2012) [1] demonstrated such an embedding. The basic idea is to introduce smooth fractal corrugations into the ring torus in order to preserve the flat metric. It is clear that embeddings of 2-dimensional Riemannian manifolds in \(\mathbb{R}^3\), while possible, is not always easy.

flat torus in R^3

The image above image (taken from [1]) shows the first four levels of corrugations for this embedding of the flat torus in \(\mathbb{R}^3\).

It is clear that even in the case of the common torus equipped with the relatively benign flat metric, embedding is non-trivial.

Going from the extrinsic picture to the intrinsic one is obvious. Moving from an intrinsic definition to an extrinsic one is more difficult, but the Nash embedding theorems show that it is generally possible (if you have enough dimensions to spare).

few questions

Here are a list of questions that I’ve had along the way.

are \(C^1\) isometric embeddings of a compact manifold unique?

No. My intuition for this is to imagine longitudinally warping a paper cylinder.

I’m curious to see if there are sufficient conditions on the metric that would guarantee a unique embedding in this case. Is imposing non-zero curvature sufficient (probably not)? I’m trying to imagine isometrically warping a paper sphere.

what are the sufficient conditions for a \(C^\infty\) embedding an \(m\)-dimensional manifold in \(\mathbb{R}^{m+1}\)?

This is essentially is trying to as “what topologies and metrics can be smoothly embedded one dimension up?” The intuition for this is that we typically imagine 2-dimensional and 3-dimensional manifolds embedded in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) respectively. What makes these objects special?

The 2-sphere embedded in \(\mathbb{R}^3\) above is one example of this.

I don’t have a good answer for this currently. This is probably something to research if I have time. I’d be very happy to talk if anyone has thoughts about this.

future work

I don’t really have anything planned here. My initial goal here was just to get a better intuition for visualizing Riemannian manifolds and to obtain some intuition for curvature of the manifolds. I didn’t expect to delve too deeply in topology and embedding theorems and it seems like I’ve just scratched the surface. Just as a note to myself, reading through a few of the things discussed in this post has motived the topic of algebraic topology; probably should look into that eventually.

References

[1] V. Borrelli, S. Jabrane, F. Lazarus, and B. Thibert, “Flat tori in three-dimensional space and convex integration,” Proceedings of the National Academy of Sciences of the United States of America, vol. 109, no. 19, pp. 7218–7223, Mar. 2012.

[2] T. Tao, “Notes on the Nash embedding theorem,” 2016. [Online]. Available: https://terrytao.wordpress.com/2016/05/11/notes-on-the-nash-embedding-theorem/. Accessed: Feb. 24, 2017.