# Introduction

In most presentations of Riemannian geometry, e.g. O’Neill (1983) and Wikipedia, the fundamental theorem of Riemannian geometry (“the miracle of Riemannian geometry”) is given: that for any semi-Riemannian manifold there is a unique torsion-free metric connection. I assume partly because of this and partly because the major application of Riemannian geometry is General Relativity, connections with torsion are given little if any attention.

It turns out we are all very familiar with a connection with torsion: the Mercator projection. Some mathematical physics texts, e.g. Nakahara (2003), allude to this but leave the details to the reader. Moreover, this connection respects the metric induced from Euclidean space.

We use SageManifolds to assist with the calculations. We hint at how this might be done more slickly in Haskell.

## A Cartographic Aside

`%matplotlib inline`

```
/Applications/SageMath/local/lib/python2.7/site-packages/traitlets/traitlets.py:770: DeprecationWarning: A parent of InlineBackend._config_changed has adopted the new @observe(change) API
clsname, change_or_name), DeprecationWarning)
```

```
import matplotlib
import numpy as np
import matplotlib.pyplot as plt
```

```
import cartopy
import cartopy.crs as ccrs
from cartopy.mpl.ticker import LongitudeFormatter, LatitudeFormatter
```

```
plt.figure(figsize=(8, 8))
ax = plt.axes(projection=cartopy.crs.Mercator())
ax.gridlines()
ax.add_feature(cartopy.feature.LAND)
ax.add_feature(cartopy.feature.COASTLINE)
plt.show()
```

We can see Greenland looks much broader at the North than in the middle. But if we use a polar projection (below) then we see this is not the case. Why then is the Mercator projection used in preference to e.g. the polar projection or the once controversial Gall-Peters – see here for more on map projections.

```
plt.figure(figsize=(8, 8))
bx = plt.axes(projection=cartopy.crs.NorthPolarStereo())
bx.set_extent([-180, 180, 90, 50], ccrs.PlateCarree())
bx.gridlines()
bx.add_feature(cartopy.feature.LAND)
bx.add_feature(cartopy.feature.COASTLINE)
plt.show()
```

## Colophon

This is written as an Jupyter notebook. In theory, it should be possible to run it assuming you have installed at least sage and Haskell. To publish it, I used

```
jupyter-nbconvert --to markdown Mercator.ipynb
pandoc -s Mercator.md -t markdown+lhs -o Mercator.lhs \
--filter pandoc-citeproc --bibliography DiffGeom.bib
BlogLiteratelyD --wplatex Mercator.lhs > Mercator.html
```

Not brilliant but good enough.

Some commands to `jupyter`

to display things nicely.

`%display latex`

`viewer3D = 'tachyon'`

# Warming Up With SageManifolds

Let us try a simple exercise: finding the connection coefficients of the Levi-Civita connection for the Euclidean metric on in polar co-ordinates.

Define the manifold.

`N = Manifold(2, 'N',r'\mathcal{N}', start_index=1)`

Define a chart and frame with Cartesian co-ordinates.

`ChartCartesianN.<x,y> = N.chart()`

`FrameCartesianN = ChartCartesianN.frame()`

Define a chart and frame with polar co-ordinates.

`ChartPolarN.<r,theta> = N.chart()`

`FramePolarN = ChartPolarN.frame()`

The standard transformation from Cartesian to polar co-ordinates.

`cartesianToPolar = ChartCartesianN.transition_map(ChartPolarN, (sqrt(x^2 + y^2), arctan(y/x)))`

`print(cartesianToPolar)`

`Change of coordinates from Chart (N, (x, y)) to Chart (N, (r, theta))`

`print(latex(cartesianToPolar.display()))`

`cartesianToPolar.set_inverse(r * cos(theta), r * sin(theta))`

```
Check of the inverse coordinate transformation:
x == x
y == y
r == abs(r)
theta == arctan(sin(theta)/cos(theta))
```

Now we define the metric to make the manifold Euclidean.

`g_e = N.metric('g_e')`

`g_e[1,1], g_e[2,2] = 1, 1`

We can display this in Cartesian co-ordinates.

`print(latex(g_e.display(FrameCartesianN)))`

And we can display it in polar co-ordinates

`print(latex(g_e.display(FramePolarN)))`

Next let us compute the Levi-Civita connection from this metric.

`nab_e = g_e.connection()`

`print(latex(nab_e))`

If we use Cartesian co-ordinates, we expect that . Only non-zero entries get printed.

`print(latex(nab_e.display(FrameCartesianN)))`

Just to be sure, we can print out all the entries.

`print(latex(nab_e[:]))`

In polar co-ordinates, we get

`print(latex(nab_e.display(FramePolarN)))`

Which we can rew-rewrite as

with all other entries being 0.

# The Sphere

We define a 2 dimensional manifold. We call it the 2-dimensional (unit) sphere but it we are going to remove a meridian to allow us to define the desired connection with torsion on it.

`S2 = Manifold(2, 'S^2', latex_name=r'\mathbb{S}^2', start_index=1)`

`print(latex(S2))`

To start off with we cover the manifold with two charts.

`polar.<th,ph> = S2.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi'); print(latex(polar))`

`mercator.<xi,ze> = S2.chart(r'xi:(-oo,oo):\xi ze:(0,2*pi):\zeta'); print(latex(mercator))`

We can now check that we have two charts.

`print(latex(S2.atlas()))`

We can then define co-ordinate frames.

`epolar = polar.frame(); print(latex(epolar))`

`emercator = mercator.frame(); print(latex(emercator))`

And define a transition map and its inverse from one frame to the other checking that they really are inverses.

`xy_to_uv = polar.transition_map(mercator, (log(tan(th/2)), ph))`

`xy_to_uv.set_inverse(2*arctan(exp(xi)), ze)`

```
Check of the inverse coordinate transformation:
th == 2*arctan(sin(1/2*th)/cos(1/2*th))
ph == ph
xi == xi
ze == ze
```

We can define the metric which is the pullback of the Euclidean metric on .

`g = S2.metric('g')`

`g[1,1], g[2,2] = 1, (sin(th))^2`

And then calculate the Levi-Civita connection defined by it.

`nab_g = g.connection()`

`print(latex(nab_g.display()))`

We know the geodesics defined by this connection are the great circles.

We can check that this connection respects the metric.

`print(latex(nab_g(g).display()))`

And that it has no torsion.

`print(latex(nab_g.torsion().display()))`

`0`

## A New Connection

Let us now define an orthonormal frame.

```
ch_basis = S2.automorphism_field()
ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
```

`e = S2.default_frame().new_frame(ch_basis, 'e')`

`print(latex(e))`

We can calculate the dual 1-forms.

`dX = S2.coframes()[2] ; print(latex(dX))`

`print(latex((dX[1], dX[2])))`

`print(latex((dX[1][:], dX[2][:])))`

In this case it is trivial to check that the frame and coframe really are orthonormal but we let `sage`

do it anyway.

`print(latex(((dX[1](e[1]).expr(), dX[1](e[2]).expr()), (dX[2](e[1]).expr(), dX[2](e[2]).expr()))))`

Let us define two vectors to be parallel if their angles to a given meridian are the same. For this to be true we must have a connection with .

`nab = S2.affine_connection('nabla', latex_name=r'\nabla')`

`nab.add_coef(e)`

Displaying the connection only gives the non-zero components.

`print(latex(nab.display(e)))`

For safety, let us check all the components explicitly.

`print(latex(nab[e,:]))`

Of course the components are *not* non-zero in other frames.

`print(latex(nab.display(epolar)))`

`print(latex(nab.display(emercator)))`

This connection also respects the metric .

`print(latex(nab(g).display()))`

Thus, since the Levi-Civita connection is unique, it *must* have torsion.

`print(latex(nab.torsion().display(e)))`

The equations for geodesics are

Explicitly for both variables in the polar co-ordinates chart.

We can check that and are solutions although `sage`

needs a bit of prompting to help it.

`t = var('t'); a = var('a')`

`print(latex(diff(a * log(tan(t/2)),t).simplify_full()))`

We can simplify this further by recalling the trignometric identity.

`print(latex(sin(2 * t).trig_expand()))`

`print(latex(diff (a / sin(t), t)))`

In the mercator co-ordinates chart this is

In other words: straight lines.

Reparametersing with we obtain

Let us draw such a curve.

`R.<t> = RealLine() ; print(R)`

`Real number line R`

`print(dim(R))`

`1`

`c = S2.curve({polar: [2*atan(exp(-t/10)), t]}, (t, -oo, +oo), name='c')`

`print(latex(c.display()))`

`c.parent()`

`c.plot(chart=polar, aspect_ratio=0.1)`

It’s not totally clear this is curved so let’s try with another example.

`d = S2.curve({polar: [2*atan(exp(-t)), t]}, (t, -oo, +oo), name='d')`

`print(latex(d.display()))`

`d.plot(chart=polar, aspect_ratio=0.2)`

Now it’s clear that a *straight* line is curved in polar co-ordinates.

But of course in Mercator co-ordinates, it is a straight line. This explains its popularity with mariners: if you draw a straight line on your chart and follow that bearing or rhumb line using a compass you will arrive at the end of the straight line. Of course, it is not the shortest path; great circles are but is much easier to navigate.

`c.plot(chart=mercator, aspect_ratio=0.1)`

`d.plot(chart=mercator, aspect_ratio=1.0)`

We can draw these curves on the sphere itself not just on its charts.

`R3 = Manifold(3, 'R^3', r'\mathbb{R}^3', start_index=1)`

`cart.<X,Y,Z> = R3.chart(); print(latex(cart))`

```
Phi = S2.diff_map(R3, {
(polar, cart): [sin(th) * cos(ph), sin(th) * sin(ph), cos(th)],
(mercator, cart): [cos(ze) / cosh(xi), sin(ze) / cosh(xi),
sinh(xi) / cosh(xi)]
},
name='Phi', latex_name=r'\Phi')
```

We can either plot using polar co-ordinates.

`graph_polar = polar.plot(chart=cart, mapping=Phi, nb_values=25, color='blue')`

`show(graph_polar, viewer=viewer3D)`

Or using Mercator co-ordinates. In either case we get the sphere (minus the prime meridian).

`graph_mercator = mercator.plot(chart=cart, mapping=Phi, nb_values=25, color='red')`

`show(graph_mercator, viewer=viewer3D)`

We can plot the curve with an angle to the meridian of

`graph_c = c.plot(mapping=Phi, max_range=40, plot_points=200, thickness=2)`

`show(graph_polar + graph_c, viewer=viewer3D)`

And we can plot the curve at angle of to the meridian.

`graph_d = d.plot(mapping=Phi, max_range=40, plot_points=200, thickness=2, color="green")`

`show(graph_polar + graph_c + graph_d, viewer=viewer3D)`

# Haskell

With automatic differentiation and symbolic numbers, symbolic differentiation is straigtforward in Haskell.

```
> import Data.Number.Symbolic
> import Numeric.AD
>
> x = var "x"
> y = var "y"
>
> test xs = jacobian ((\x -> [x]) . f) xs
> where
> f [x, y] = sqrt $ x^2 + y^2
```

```
ghci> test [1, 1]
[[0.7071067811865475,0.7071067811865475]]
ghci> test [x, y]
[[x/(2.0*sqrt (x*x+y*y))+x/(2.0*sqrt (x*x+y*y)),y/(2.0*sqrt (x*x+y*y))+y/(2.0*sqrt (x*x+y*y))]]
```

Anyone wishing to take on the task of producing a Haskell version of sagemanifolds is advised to look here before embarking on the task.

# Appendix A: Conformal Equivalence

Agricola and Thier (2004) shows that the geodesics of the Levi-Civita connection of a conformally equivalent metric are the geodesics of a connection with vectorial torsion. Let’s put some but not all the flesh on the bones.

The **Koszul formula** (see e.g. (O’Neill 1983)) characterizes the Levi-Civita connection

Being more explicit about the metric, this can be re-written as

Let be the Levi-Civita connection for the metric where . Following [Gadea2010] and substituting into the Koszul formula and then applying the product rule

Where as usual the vector field, for , is defined via .

Let’s try an example.

`nab_tilde = S2.affine_connection('nabla_t', r'\tilde_{\nabla}')`

`f = S2.scalar_field(-ln(sin(th)), name='f')`

```
for i in S2.irange():
for j in S2.irange():
for k in S2.irange():
nab_tilde.add_coef()[k,i,j] = \
nab_g(polar.frame()[i])(polar.frame()[j])(polar.coframe()[k]) + \
polar.frame()[i](f) * polar.frame()[j](polar.coframe()[k]) + \
polar.frame()[j](f) * polar.frame()[i](polar.coframe()[k]) + \
g(polar.frame()[i], polar.frame()[j]) * \
polar.frame()[1](polar.coframe()[k]) * cos(th) / sin(th)
```

`print(latex(nab_tilde.display()))`

`print(latex(nab_tilde.torsion().display()))`

`0`

`g_tilde = exp(2 * f) * g`

`print(latex(g_tilde.parent()))`

`print(latex(g_tilde[:]))`

`nab_g_tilde = g_tilde.connection()`

`print(latex(nab_g_tilde.display()))`

It’s not clear (to me at any rate) what the solutions are to the geodesic equations despite the guarantees of Agricola and Thier (2004). But let’s try a different chart.

`print(latex(nab_g_tilde[emercator,:]))`

In this chart, the geodesics are clearly straight lines as we would hope.

# References

Agricola, Ilka, and Christian Thier. 2004. “The geodesics of metric connections with vectorial torsion.” *Annals of Global Analysis and Geometry* 26 (4): 321–32. doi:10.1023/B:AGAG.0000047509.63818.4f.

Nakahara, M. 2003. “Geometry, Topology and Physics.” *Text* 822: 173–204. doi:10.1007/978-3-642-14700-5.

O’Neill, B. 1983. *Semi-Riemannian Geometry with Applications to Relativity, 103*. Pure and Applied Mathematics. Elsevier Science. https://books.google.com.au/books?id=CGk1eRSjFIIC.