# Illustration of Jacobi Fields

This tutorial illustrates the usage of Jacobi Fields within `Manopt.jl`

. For this tutorial you should be familiar with the basic terminology on a manifold like the exponential and logarithmic map as well as shortest geodesics.

We first initialize the packages we need

`using Colors, Manopt, Manifolds, PlutoUI`

and we define some colors from Paul Tol

```
begin
black = RGBA{Float64}(colorant"#000000")
TolVibrantMagenta = RGBA{Float64}(colorant"#EE3377")
TolVibrantOrange = RGBA{Float64}(colorant"#EE7733")
TolVibrantCyan = RGBA{Float64}(colorant"#33BBEE")
TolVibrantTeal = RGBA{Float64}(colorant"#009988")
end;
```

And setup our graphics paths

```
begin
localpath = join(splitpath(@__FILE__)[1:(end - 1)], "/") # files folder
image_prefix = localpath * "/jacobi_fields"
@info image_prefix
render_asy = false # on CI or when you do not have asymptote, this should be false
end;
```

Assume we have two points on the equator of the Sphere $\mathcal M = \mathbb S^2$

`M = Sphere(2)`

`Sphere(2, ℝ)`

`p, q = [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0]]`

```
2-element Vector{Vector{Float64}}:
[1.0, 0.0, 0.0]
[0.0, 1.0, 0.0]
```

their connecting shortest geodesic (sampled at `100`

points)

`geodesic_curve = shortest_geodesic(M, p, q, [0:0.1:1.0...]);`

Is just a curve along the equator

```
render_asy && asymptote_export_S2_signals(
image_prefix * "/jacobi_geodesic.asy";
curves=[geodesic_curve],
points=[[p, q]],
colors=Dict(:curves => [black], :points => [TolVibrantOrange]),
dot_size=3.5,
line_width=0.75,
camera_position=(1.0, 1.0, 0.5),
);
```

`render_asy && render_asymptote(image_prefix * "/jacobi_geodesic.asy"; render=2);`

`PlutoUI.LocalResource(image_prefix * "/jacobi_geodesic.png")`

where $p$ is on the left and $q$ on the right. We know solve the following task:

Given a direction $X ∈ T_p\mathcal M$, for example

`X = [0.0, 0.4, 0.5]`

```
3-element Vector{Float64}:
0.0
0.4
0.5
```

we move the start point $x$ into, how does any point on the geodesic move?

Or mathematically: Compute $D_p g(t; p,q)$ for some fixed $t∈[0,1]$ and a given direction $X_p$. Of course two cases are quite easy: For $t=0$ we are in $x$ and how $x$ “moves” is already known, so $D_x g(0;p,q) = X$. On the other side, for $t=1$, $g(1; p,q) = q$ which is fixed, so $D_p g(1; p,q)$ is the zero tangent vector (in $T_q\mathcal M$).

For all other cases we employ a `jacobi_field`

, which is a (tangent) vector field along the shortest geodesic given as follows: The *geodesic variation* $\Gamma_{g,X}(s,t)$ is defined for some $\varepsilon > 0$ as

$$\Gamma_{g,X}(s,t):=\exp{\gamma_{p,X}(s)}[t\log_{g(s;p,X)}p],\qquad s∈(-\varepsilon,\varepsilon),\ t∈[0,1].$$

Intuitively we make a small step $s$ into direction $ξ$ using the geodesic $g(⋅; p,X)$ and from $r=g(s; p,X)$ we follow (in $t$) the geodesic $g(⋅; r,q)$. The corresponding Jacobi field $J_{g,X}$ along $g(⋅; p,q)$ is given by

$$J_{g,X}(t):=\frac{D}{\partial s}\Gamma_{g,X}(s,t)\Bigl\rvert_{s=0}$$

which is an ODE and we know the boundary conditions $J_{g,X}(0)=X$ and $J_{g,X}(t) = 0$. In symmetric spaces we can compute the solution, since the system of ODEs decouples, see for example [doCarmo1992], Chapter 4.2. Within `Manopt.jl`

this is implemented as `jacobi_field(M,p,q,t,X[,β])`

, where the optional parameter (function) $β$ specifies, which Jacobi field we want to evaluate and the one used here is the default.

We can hence evaluate that on the points on the geodesic at

`T = [0:0.1:1.0...];`

namely

`r = shortest_geodesic(M, p, q, T);`

The tangent vector now moves as a differential along the geodesic as

`W = jacobi_field.(Ref(M), Ref(p), Ref(q), T, Ref(X));`

Which can also be called using `differential_geodesic_startpoint`

. We can add to the image above by creating extended tangent vectors the include their base points

`V = [Tuple([a, b]) for (a, b) in zip(r, W)];`

to add the vectors as one further set to the Asymptote export.

```
render_asy && asymptote_export_S2_signals(
image_prefix * "/jacobi_geodesic_diff_start.asy";
curves=[geodesic_curve],
points=[[p, q], r],
tangent_vectors=[V],
colors=Dict(
:curves => [black],
:points => [TolVibrantOrange, TolVibrantCyan],
:tvectors => [TolVibrantCyan],
),
dot_sizes=[3.5, 2.0],
line_width=0.75,
camera_position=(1.0, 1.0, 0.5),
);
```

`render_asy && render_asymptote(image_prefix * "/jacobi_geodesic_diff_start.asy"; render=2);`

`PlutoUI.LocalResource(image_prefix * "/jacobi_geodesic_diff_start.png")`

The interpretation is as follows: If an infinitesimal change of the start point in direction $X$ would happen, the infinitesimal change of a point along the line would change as indicated.

Note that each new vector is a tangent vector to its point (up to a small numerical tolerance), so the blue vectors are not just “shifted and scaled versions” of $X$.

`all([is_vector(M, a[1], a[2]; atol=1e-15) for a in V])`

`true`

If we further move the end point, too, we can derive that Differential in direction

```
begin
Y = [0.2, 0.0, -0.5]
W2 = differential_geodesic_endpoint.(Ref(M), Ref(p), Ref(q), T, Ref(Y))
V2 = [Tuple([a, b]) for (a, b) in zip(r, W2)]
end;
```

and we can combine both vector fields we obtained

`V3 = [Tuple([a, b]) for (a, b) in zip(r, W2 + W)];`

```
render_asy && asymptote_export_S2_signals(
image_prefix * "/jacobi_geodesic_complete.asy";
curves=[geodesic_curve],
points=[[p, q], r],
tangent_vectors=[V, V2, V3],
colors=Dict(
:curves => [black],
:points => [TolVibrantOrange, TolVibrantCyan],
:tvectors => [TolVibrantCyan, TolVibrantMagenta, TolVibrantTeal],
),
dot_sizes=[3.5, 2.0],
line_width=0.75,
camera_position=(1.0, 1.0, 0.0),
);
```

`render_asy && render_asymptote(image_prefix * "/jacobi_geodesic_complete.asy"; render=2);`

`PlutoUI.LocalResource(image_prefix * "/jacobi_geodesic_complete.png")`

Here the first vector field is still in blue, the second is in magenta, and their combined effect is in teal. Sure as a differential this does not make much sense, maybe as an infinitesimal movement of both start and end point cmobined.

## Literature

doCarmo1992

do Carmo, Manfredo

Riemannian Geometry, Birkhäuser Boston, 1992, ISBN: 0-8176-3490-8.