### Curvature and Geodesics on a Torus

Geodesics on a torus [image from Jantzen, Robert T., 2021].

We take a look at the curvature on a torus, and the various forms that geodesics can have. These are compared to the geodesics on a “flat torus”.

Toroidal-Poloidal Coordinates

The position on a torus may be specified by the toroidal and poloidal coordinates. The toroidal component (${\lambda}$) is the angle following a large circle around the torus. It is analogous to the longitude, varying along the parallels. The poloidal coordinate (${\theta}$) varies along a smaller circle around the surface. Changing ${\theta}$ is like altering the latitude. We take ${\lambda\in[0,2\pi)}$ and ${-\pi \le \theta<\pi}$.

We can map the torus onto a plane by cutting it along the innermost parallel and opening it out flat. Obviously, this involves stretching; the mapping is not an isometry. We note some particular parallels: the outer equator (OE) is the longest parallel; the inner equator (IE) is the shortest; the north pole circle (NPC) and south pole circle (SPC) are the parallel at the top and bottom of the torus. Note that in the figure above, the left and right sides (${\lambda=0}$ and ${\lambda=2\pi}$) are to be identified and also the upper and lower boundaries (${\theta=-\pi}$ and ${\theta=\pi}$).

Metric and Curvature

The cartesian coordinates for a point on the torus with toroidal/poloidal coordinates ${(\lambda,\theta)}$ are given by

$\displaystyle \begin{array}{rcl} x &=& (R + r \cos\theta )\cos\lambda \\ y &=& (R + r \cos\theta )\sin\lambda \\ z &=& r \sin\theta \,. \end{array}$

The position vector of a point on the surface is

$\displaystyle \mathbf{x}(\lambda,\phi) = \langle (R + r \cos\theta )\cos\lambda, (R + r \cos\theta)\sin\lambda, r \sin\theta \rangle$

The tangent vectors in the toroidal and poloidal directions are

$\displaystyle \begin{array}{rcl} \mathbf{e}_\lambda = \frac{\partial\mathbf{x}}{\partial\lambda} &=& (-(R + r \cos\theta )\sin\lambda, (R + r \cos\theta)\cos\lambda, 0 ) \\ \mathbf{e}_\theta = \frac{\partial\mathbf{x}}{\partial\theta} &=& ( r \sin\theta\cos\lambda, r \cos\theta\sin\lambda, 0 ) \,. \end{array}$

From these, the metric coefficients ${g_{ij} = \mathbf{e}_i\mathbf{\cdot e}_j}$ follow:

$\displaystyle g_{11} = (R + r \cos\theta )^2 \qquad g_{12} = g_{21} = 0 \qquad g_{22} = r^2 \,.$

We can thus write down the equation for a line element:

$\displaystyle \mathrm{d}s^2 = g_{\mu\nu} \mathrm{d}x^\mu \mathrm{d}x^\nu = (R + r \cos\theta )\mathrm{d}\lambda + r^2\mathrm{d}\theta \,.$

It is now straightforward to compute the Christoffel symbols and the Riemann curvature tensor ${R^{\kappa}_{\lambda\mu\nu}}$, contract it to get the Ricci tensor ${R_{\mu\nu}}$ and Ricci curvature ${R}$, from which the gaussian curvature follows. All of this can be done with open-source software.

For a two-dimensional surface, the symmetries of the Riemann curvature tensor imply that there is just one independent component, which we take to the ${R_{1212}}$. This turns out to be

$\displaystyle R_{1212} = - {r\cos\theta}{(R + r\cos\theta)}$

and it is related to the gaussian curvature by ${K = - R_{1212}/ \det{g}}$. As a consequence,

$\displaystyle K = \frac{\cos\theta} {r(R+r\cos\theta)} \,.$

Since we assume that ${r < R}$, the gaussian curvature is positive for ${|\theta|<\pi/2}$ and negative for ${-\pi<\theta<-\pi/2}$ and ${\pi/2<\theta<\pi}$. This makes sense: the curvature is positive on the outside of the torus and negative on the inside.

We can write the equations for the geodesics. In the general form, they are

$\displaystyle \frac{\mathrm{d}^2 x^\alpha}{\mathrm{d}s^2} + \Gamma^\alpha_{\mu\nu} x^\mu x^\nu = 0 \,.$

Given an initial position and slope, these can be solved, analytically or numerically. The only parallels that are geodesics are the outer equator (OE) and the inner equator (IE). All meridians (${\lambda}$ constant) are geodesics.

Schematic diagram of geodesics on a torus. All curves start from ${(\lambda, \theta) = (0, 0)}$. OE: Outer Equator (${\theta=0}$). IE: Inner Equator (${\theta=\pi}$). NPC: North Pole Circle (${\theta=\pi/2}$). SPC: South Pole Circle (${\theta=-\pi/2}$).}

We show some typical geodesics in the above figure;  all start from the position ${\lambda=\theta=0}$. There are two families of generic curves. For the curves that do not pass through the central hole, the poloidal angle is bounded, ${|\theta|\le\theta_b<\pi}$. These are represented by the blue curve. Geodesics that do pass through the central hole are represented by the red curve; for these, the poloidal angle varies monotonically. The families are separated by a special geodesic that approaches the inner equator asymptotically (thick black curve).

The Flat Torus

If we assume that ${R}$ is large and ${r=O(1)}$, the curvature ${K \approx \cos\theta/(rR)}$ is small. Keeping the topological form of the torus but setting the curvature to zero, we get the flat torus. It is clear that all geodesics on the flat torus are straight lines. All parallels and all meridians are geodesics (horizontal and vertical lines on the diagram). The geodesics through the origin are ${\theta = m\lambda}$. If ${m=r/s}$ is rational, then ${r\lambda = s\theta}$ and the geodesic closes on itself — it is periodic. If ${m}$ is irrational, the geodesic covers the torus densely. The periodic curves are the toroidal knots. They can be identified by the numbers ${(r,s)}$, which give the number of loops around the “long” and “short” circuits of the torus.

Sources

${\bullet}$ Kyle Celano, Vincent E. Coll & Jeff Dodd (2022): Why Curves Curve: The Geodesics on the Torus, Mathematics Magazine, 95:3, 230–239, DOI: 10.1080/0025570X.2022.2055349

${\bullet}$ Jantzen, Robert T., 2021: Geodesics on the Torus and other Surfaces of Revolution. ArXiv.

${\bullet}$ William Schulz, Differential Geometry attacks the torus.  PDF.