Invariant Metric and Geodesics

The Euclidean metric is not preserved by automorphisms of the Lobachevsky half-plane. Instead, one has only a weaker property of conformality. However, it is possible to find such a metric on the Lobachevsky half-plane that Möbius transformations will be isometries. Similarly, in Chapter ??, we described a variety of distances and lengths and many of them had conformal properties with respect to SL₂(ℝ) action. However, it is worth finding a metric which is preserved by Möbius transformations. We will now proceed do this, closely following [165].

Our consideration will be based on equidistant orbits, which physically correspond to wavefronts with a constant velocity. For example, if a stone is dropped into a pond, the resulting ripples are waves which travel the same distance from the drop point, assuming a constant wave velocity. An alternative description to wavefronts uses rays—the paths along which waves travel, i.e. the geodesics in the case of a constant velocity. The duality between wavefronts and rays is provided by Huygens’ principle —see [11]*§ 46.

Geodesics also play a central role in differential geometry, generalising the notion of a straight line. They are closely related to metrics: a geodesic is often defined as a curve between two points with an extremum of length. As a consequence, the metric is additive along geodesics.

9.1 Metrics, Curves’ Lengths and Extrema

Definition 1 A metric on a set X is a function d: X × X → ℝ^+ such that

d(x, y) ≥ 0 (positivity),
d(x, y)=0 if and only if x=y (non-degeneracy),
d(x, y)=d(y, x) (symmetry) and
d(x,y)≤ d(x,z)+d(z,y) (the triangle inequality ),

for all x, y, z ∈ X.

Although adequate in many cases, the definition does not cover all metrics of interest. Examples include the non-symmetric lengths from Section 7.2 or distances (5) in the Minkowski space with the reverse triangle inequality d(x,y)≥ d(x,z)+d(z,y).

Recall the established procedure of constructing geodesics in Riemannian geometry (two-dimensional case) from [337]*§ 7:

Exercise 2 Let the quadratic form (1) be SL₂(ℝ)-invariant. Show that the above procedure leads to an SL₂(ℝ)-invariant metric.

We recall from Section 3.7 that an isotropy subgroup H fixing the hypercomplex unit ι under the action of (4) is K (3), N′ (6) and A′ (5) in the corresponding EPH cases. We will refer to H-action as EPH rotation around ι. For an SL₂(ℝ) invariant metric, the orbits of H will be equidistant points from ι, giving some indication what the metric should be. However, this does not determine the metric entirely since there is freedom in assigning values to the orbits.

Lemma 3 The invariant infinitesimal metric in EPH cases is

d s²=

du²−σ dv²

v²

, (3)

where σ=−1,0,1 respectively.

Proof. In order to calculate the infinitesimal metric, consider the subgroups H of Möbius transformations which fix ι. Denote an element of these rotations by E_σ. We require an isometry, so

A metric is invariant under the above rotations if it is preserved under the linear transformation

To calculate the metric at an arbitrary point w=u+iv, we map w to ι by an affine Möbius transformation, which acts transitively on the upper half-plane

Hence, there is a factor of 1/v², so the resulting metric is ds²=du²−σ dv²/v².

Corollary 4 With the above notation, for an arbitrary curve Γ,

length(Γ)=

∫

(du²−σ dv²)

. (5)

It is invariant under Möbius transformations of the upper half-plane.

The standard tools for finding geodesics for a given Riemannian metric are the Euler–Lagrange equations—see [337]*§ 7.1. For the metric (3), they take the form

where γ is a smooth curve γ(t)=(γ₁(t),γ₂(t)) and t ∈ (a,b). This general approach can be used in the two non-degenerate cases (elliptic and hyperbolic) and produces curves with the minimum or maximum lengths, respectively. However, the SL₂(ℝ)-invariance of the metric allows us to use more elegant methods in this case. For example, in the Lobachevsky half-plane, the solutions are well known—semicircles orthogonal to the real axes or vertical lines, cf. Fig. 9.1(a) and [26]*Ch. 15.

Exercise 5 For the elliptic space (σ=−1):

Write a parametric equation of a circle orthogonal to the real line and check that the curve satisfies the Euler–Lagrange equations (6).
Geodesics passing the imaginary unit i are transverse circles to K-orbits from Fig. 1.2 and Exercise 4 with the equation, cf. Fig. 9.3(E)
(u² +v²) −2t u −1 =0, where t ∈ ℝ. (7)
The Möbius invariant metric is
m(z,w)=sinh⁻¹
⎪
⎪ z−w ⎪
⎪ _e

2 √

ℑ[z]ℑ[w]

, (8)

where ℑ[z] is the imaginary part of a complex number z and | z |_e²=u²+v². Hint: One can directly or, by using CAS, verify that this is a Möbius-invariant expression. Thus, we can transform z and w to i and a point on the imaginary axis by a suitable Möbius transformation without changing the metric. The shortest curve in the Riemannian metric (3) is the vertical line, that is, du=0. For a segment of the vertical line, the expression (8) can easily be evaluated. See also [24]*Thm. 7.2.1 for detailed proof and a number of alternative expressions. ⋄

Exercise 6 Show, similarly, that, in the hyperbolic case (σ=1):

There are two families of solutions passing the double unit є, one space-like (Fig. 9.3(H_S)) and one time-like (Fig. 9.3(H_T)), cf. Section 8.4:
(u²−v²) −2tu +1=0, where ⎪
⎪ t ⎪
⎪ >1 or ⎪
⎪ t ⎪
⎪ <1. (9)

The space-like solutions are obtained by rotation of the vertical axis and the time-like solution is the -image of the cycle (1,0,0,1)—cf. Fig. 3.1. They also consist of positive and negative cycles, respectively, which are orthogonal to the real axis.

The respective metric in these two cases is:

d(z,w)=

⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩

2 sin⁻¹

⎪
⎪

z−w

⎪
⎪

√

ℑ[z]ℑ[w]

when z−w is time-like

2 sinh⁻¹

⎪
⎪

z−w

⎪
⎪

√

ℑ[z]ℑ[w]

when z−w is space-like,

(10)

where ℑ[z] is the imaginary part of a double number z and | z |_h²=u²−v². Hint: The hint from the previous exercise can be used again here with some modification to space-/time-like curves and by replacing the minimum of the possible curves’ lengths by the maximum. ⋄

The same geodesic equations can be obtained by Beltrami’s method—see [42]*§ 8.1. However, the parabolic case presents yet another disappointment.

Exercise 7 Show that, for the parabolic case (σ=0):

The only solution of the Euler–Lagrange equations (6) are vertical lines, as in [339]*§ 3, which are again orthogonal to the real axes.
Vertical lines minimise the curves’ length between two points w₁ and w₂. See Fig. 9.1(b) for an example of a blue curve with its length tending to zero value of the infimum. The respective “geodesics” passing the dual unit ε are
u²−2tu=0. (11)

Similarly, a length of the red curve can be arbitrarily large if the horizontal path is close enough to the real axis.
The only SL₂(ℝ)-invariant metric obtained from the above extremal consideration is either identically equal to 0 or infinity.

There is a similarity between all three cases. For example, we can uniformly write equations (7), (9) and (11) of geodesics through ι as

However, the triviality of the parabolic invariant metric is awkward and we go on to study further the algebraic and geometric invariants to find a more adequate answer.

9.2 Invariant Metric

We seek all real-valued functions f of two points on the half-plane which are invariant under the Möbius action

which can be shown by a simple direct calculation (for CAS, see Exercise 3). Recall that | z |_σ²=u²−σ v² by analogy with the distance d_σ,σ (5) in EPH geometries and [339]*App. C. In order to describe other invariant functions we will need the following definition.

Definition 8 A function f : X × X → ℝ^+ is called a monotonous metric if f(Γ(0), Γ(t)) is a continuous monotonically-increasing function of t, where Γ : [0,1) → X is a smooth curve with Γ(0)=z₀ that intersects all equidistant orbits of z₀ exactly once.

Exercise 9 Check the following:

The function F(z,w) (12) is monotonous.
If h is a monotonically-increasing, continuous, real function, then f(z, w)=h ∘ F(z,w) is a monotonous SL₂(ℝ)-invariant function.

Theorem 10 A monotonous function f(z,w) is invariant under g ∈ SL₂(ℝ) if and only if there exists a monotonically-increasing, continuous, real function h such that f(z, w)=h ∘ F(z,w).

Proof. Because of the Exercise 2, we show the necessity only. Suppose there exists another function with such a property, say, H(z,w). Due to invariance under SL₂(ℝ), this can be viewed as a function in one variable if we apply r⁻¹ (cf. (4)) which sends z to ι and w to r⁻¹(w). Now, by considering a fixed smooth curve Γ from Definition 8, we can completely define H(z,w) as a function of a single real variable h(t)=H(i, Γ(t)) and, similarly for F(z,w),

where h and f are both continuous and monotonically-increasing since they represent metrics. Hence, the inverse f⁻¹ exists everywhere by the inverse function theorem. So,

Note that hf⁻¹ is monotonic, since it is the composition of two monotonically-increasing functions, and this ends the proof.

Remark 11 The above proof carries over to a more general theorem which states: If there exist two monotonous functions F(u,v) and H(u,v) which are invariant under a transitive action of a group G, then there exists a monotonically-increasing real function h such that H(z,w)=h ∘ F(z,w).

As discussed in the previous section, in elliptic and hyperbolic geometries the function h from above is either sinh⁻¹t or sin⁻¹t (8) (10). Hence, it is reasonable to try inverse trigonometric and hyperbolic functions in the intermediate parabolic case as well.

Remark 12 The above result sheds light on the possibilities we have—we can either

“label” the equidistant orbits with numbers, i.e choose a function h which will then determine the geodesic, or
choose a geodesic which will then determine h.

These two approaches are reflected in the next two sections.

9.3 Geodesics: Additivity of Metric

As pointed out earlier, there might not be a metric function which satisfies all the traditional properties. However, we still need the key ones, and we therefore make the following definition:

Definition 13 Geodesics for a metric d are smooth curves along which d is additive, that is d(x,y)+d(y,z)=d(x,z), for any three point, of the curve, such that y is between x and z.

Remark 14 It is important that this definition is relevant in all EPH cases, i.e. in the elliptic and hyperbolic cases it would produce the well-known geodesics defined by the extremality condition.

Exercise 15 Let γ be a geodesic for a metric d. Write the differential equation for γ in term of d. Hint: Consider d(w, w′) as a real function in four variables, say f(u, v, u′, v′). Then write the infinitesimal version of the additivity condition and obtain the equation

δ v

δ u

−f′₃(u, v, u′,v′) +f′₃(u′,v′,u′,v′)

f′₄(u, v, u′,v′) −f′₄(u′,v′,u′,v′)

, (15)

where f′_n stands for the partial derivative of f with respect to the n-th variable. ⋄

where the elliptic, parabolic and hyperbolic inverse sine functions are (see [190] [129])

Note that σ_r is independent of σ although it takes the same three values, similar to the different signatures of point and cycle spaces introduced in Chapter ??. It is used to denote the possible sub-cases within the parabolic geometry alone.

Exercise 16 Check that:

For u=0, v=1 and the metric d_{p,σ_r} (16), Equation (15) is:

δ v

δ u

=
2v

u

−
√

⎪
⎪ σ_r u²+4v ⎪
⎪

u

.
The geodesics through ι for the metric d_{p,σ_r} (16) are parabolas:
(σ_r+4t²)u²−8tu−4v+4=0. (18)

Let us verify which properties from Definition 1 are satisfied by the invariant metric derived from (16). Two of the four properties hold—it is clearly symmetric and positive for every two points. However, the metric of any point to a point on the same vertical line is zero, so d(z, w)=0 does not imply z=w. This can be overcome by introducing a different metric function just for the points on the vertical lines—see [339]*§ 3. Note that we still have d(z,z)=0 for all z.

The triangle inequality holds only in the elliptic point space, whereas, in the hyperbolic point space, we have the reverse situation: d(w₁,w₂)≥ d(w₁,z)+d(z,w₂). There is an intermediate situation in the parabolic point space:

Lemma 17 Take any SL₂(ℝ)-invariant metric function and take two points w₁, w₂ and the geodesic (in the sense of Definition 13) through the points. Consider the strip ℜ[w₁]<u<ℜ[w₂] and take a point z in it. Then, the geodesic divides the strip into two regions where d(w₁,w₂)≤ d(w₁,z)+d(z,w₂) and where d(w₁,w₂)≥ d(w₁,z)+d(z,w₂).

Proof. The only possible invariant metric function in parabolic geometry is of the form d(z, w)=h ∘ |ℜ[z−w] |/2√ℑ[z]ℑ[w], where h is a monotonically-increasing, continuous, real function by Theorem 10. Fix two points w₁, w₂ and the geodesic though them. Now consider some point z=a+ib in the strip. The metric function is additive along a geodesic, so d(w₁, w₂)= d(w₁, w(a))+d(w(a),w₂), where w(a) is a point on the geodesic with real part equal to a. However, if ℑ[w(a)]<b, then d(w₁, w(a))>d(w₁, z) and d(w(a), w₂)>d(z, w₂), which implies d(w₁,w₂)> d(w₁,z)+d(z,w₂). Similarly, if ℑ[w(a)]>b, then d(w₁,w₂)< d(w₁,z)+d(z,w₂).

Remark 18 The reason for the ease with which the result is obtained is the fact that the metric function is additive along the geodesics. This justifies the Definition 13 of geodesics, in terms of additivity.

To illustrate these ideas, look at the region where the converse of the triangular inequality holds for d(z,w)_{σ_r}=sin_{σ_r}⁻¹ |ℜ[z−w] |/2√ℑ[z]ℑ[w], shaded red in Fig. 9.2. It is enclosed by two parabolas both of the form (σ_r +4t²)u²−8tu−4v+4=0 (which is the general equation of geodesics) and both passing though the two fixed points. The parabolas arise from taking both signs of the root, when solving the quadratic equation to find t. Segments of these parabolas, which bound the red region, are of different types—one of them is between points w₁ and w₂, the second joins these points with infinity.

9.4 Geometric Invariants

In the previous section, we defined an invariant metric and derived the respective geodesics. Now, we will proceed in the opposite direction. As we discussed in Exercise 7, the parabolic invariant metric obtained from the extremality condition is trivial. We work out an invariant metric from the Riemannian metric and predefined geodesics. It is schematically depicted, cf. (14), by

A minimal requirement for the family of geodesics is that they should form an invariant subset of an invariant class of curves with no more than one curve joining every two points. Thus, if we are looking for SL₂(ℝ)-invariant metrics it is natural to ask whether geodesic are cycles. An invariant subset of cycles may be characterised by an invariant algebraic condition, e.g. orthogonality. However, the ordinary orthogonality is already fulfilled for the trivial geodesics from Exercise 7, so, instead, we will try f-orthogonality to the real axes, Definition 34. Recall that a cycle is f-orthogonal to the real axes if the real axes inverted in a cycle are orthogonal (in the usual sense) to the real axes.

Exercise 19 Check that a parabola ku²−2lu−2nv+m=0 is σ_r-f-orthogonal to the real line if l²+σ_r n²−mk=0, i.e. it is a (−σ_r)-zero radius cycle.

As a starting point, consider the cycles that pass though ι. It is enough to specify only one such f-orthogonal cycle—the rest will be obtained by Möbius transformations fixing ι, i.e parabolic rotations. Within these constraints there are three different families of parabolas determined by the value σ_r.

Exercise 20 Check that the main parabolas passing ε

σ_r u²−4v+4=0, (20)

where σ_r=−1, 0, 1, are f-orthogonal to the real line. Their rotations by an element of N′ are parabolas (18).

Note that these are exactly the same geodesics obtained in (18). Hence, we already know what the metric function has to be. However, it is instructive to make the calculation from scratch since it does not involve anything from the previous section and is, in a way, more elementary and intuitive.

Exercise 21 Follow these steps to calculate the invariant metric:

Calculate the metric from ε to a point on the main parabola (20). Hint: Depending on whether the discriminant of the denominator in the integral (2) is positive, zero or negative, the results are trigonometric, rationals or hyperbolic, respectively:

∫

σt²+1

⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩

4log

2+u

2−u

if σ_r=−1,

if σ_r=0,

tan⁻¹

if σ_r=1.

This is another example of EPH classification. ⋄

For a generic point (u,v), find the N′ rotation which puts the point on the main parabola (20).
For two given points w and w′, combine the Möbius transformation g such that g:w↦ ε with the N′-rotation which puts g(w′) on the main parabola.
Deduce from the previous items the invariant metric from ε to (u,v) and check that it is a multiple of (16).
The invariant parabolic metric for σ_r=1 is equal to the angle between the tangents to the geodesic at its endpoints.

We meet an example of the splitting of the parabolic geometry into three different sub-cases, this will followed by three types of Cayley transform in Section 10.3 and Fig. 10.3. The respective geodesics and equidistant orbits have been drawn in Fig. 9.3. There is one more gradual transformation between the different geometries. We can see the transitions from the elliptic case to P_e, then to P_p, to P_h, to hyperbolic light-like and, finally, to space-like. To link it back, we observe a similarity between the final space-like case and the initial elliptic one.

Exercise 22 Show that all parabolic geodesics from ε for a given value of σ_r touch a certain parabola. This parabola can be called the horizon because geodesic rays will never reach points outside it. Note the similar effect for space-like and time-like geodesics in the hyperbolic case. Hint: This fact is a parabolic counterpart of Exercise 29. ⋄

There is one more useful parallel between all the geometries. In Lobachevsky and Minkowski geometries, the centres of geodesics lie on the real axes. In parabolic geometry, the respective σ_r-foci (see Definition 2) of σ_r-geodesic parabolas lie on the real axes. This fact is due to the relations between f-orthogonality and foci, cf. Proposition 40.

9.5 Invariant Metric and Cross-Ratio

A very elegant presentation of the Möbius-invariant metric in the Lobachevsky half-plane is based on the cross-ratio (14)—see [26]*§ 15.2.

Let w₁ and w₂ be two different points of the Lobachevsky half-plane. Draw a selfadjoint circle (i.e. orthogonal to the real line) which passes through w₁ and w₂. Let z₁ and z₂ be the points of intersection of the circle and the real line, cf. Fig. 9.1(a). From Exercise 3, a cross-ratio of four concyclic points is real, thus we define the function of two points

Exercise 23 Show the following properties of ρ:

It is Möbius-invariant, i.e. ρ(w₁,w₂)=ρ(g· w₁, g· w₂) for any g∈SL₂(ℝ). Hint: Use the fact that selfadjoint cycles form an invariant family and the Möbius-invariance of the cross-ratio, cf. Exercise 4. ⋄
It is additive, that is, for any three different points w₁, w₂, w₃ on the arc of a selfadjoint circle we have ρ(w₁,w₃)=ρ(w₁,w₂)+ρ(w₂,w₃). Hint: Use the cancellation formula (18). ⋄
It coincides with the Lobachevsky metric (8). Hint: Evaluate that ρ(i,i a)= loga for any real a>1. Thus, ρ(i,i a) is the Lobachevsky metric between i and i a. Then, apply the Möbius-invariance of ρ and the Lobachevsky metric to extend the identity for any pair of points. ⋄

This approach to the invariant metric cannot be transferred to the parabolic and hyperbolic cases in a straightforward manner for geometric reasons. As we can see from Fig. 9.3, there are certain types of geodesics which do not meet the real line. However, the case of e-geodesics in ℝ_p, which is the closest relative of the Lobachevsky half-plane, offers such a possibility.

Exercise 24 Check that:

[−2,ε,w,2]=(u+2)/(2−u), where w=u+ε(1−u²/4), u>0 is the point of the e-geodesics (main parabola) (20) passing ε and 2.
Let z₁ and z₂ be the intersections of the real line and the e-geodesic (18) passing two different points w₁ and w₂ in the upper half-plane, then ρ(w₁,w₂)=4log[z₁,w₁,w₂,z₂] is the invariant metric. Hint: Note that ρ(ε,z)=4log[−2,ε,z,2] gives the metric calculated in Exercise 1 and use the Möbius-invariance of the cross-ratio. ⋄
In the case of p-geodesics in ℝ_p, find a fixed parabola C in the upper half-plane which replaces the real line in the following sense. The cross-ratio of two points and two intersections of C with p-geodesics though the points produces an invariant metric. Find also the corresponding parabola for h-geodesics.

This is only a partial success in transferring of the elliptic theory to dual numbers. A more unified treatment for all EPH cases can be obtained from the projective cross-ratio [52]—see Section 4.5 for corresponding definitions and results. In addition, we define the map P(x,y)= x/y on the subset of ℙ¹(ℝ_σ) consisting of vectors (x,y)∈ℝ_σ² such that y is not a zero divisor. It is a left inverse of the map S(z)=(z,1) from Section 4.5.

Exercise 25 [52] Let w₁, w₂∈ ℙ¹(ℝ_σ) be two essentially distinct points .

Let w₁ and w₂ be essentially distinct, express a generic invariant metric in ℙ¹(ℝ_σ) through [w₁,w₂,w₂,w₁] using the Möbius-invariance of the projective cross-ratio and the method from Section 9.2.
For ℝ_σ being complex or double numbers, let w₁ and w₂ be in the domain of P and z_i=P(w_i), i=1, 2. Show that, cf. Fig. 9.1(a),
[w₁,w₁,w₂,w₂] =S ⎛
⎜
⎜
⎝
⎪
⎪ z₁−z₂ ⎪
⎪ _σ²

4ℑ[z₁]ℑ[z₂]

⎞
⎟
⎟
⎠ . (21)

Deduce a generic Möbius-invariant metric on ℝ_σ based on (12).

To obtain the respective notion of geodesics in ℙ¹(ℝ_σ) from the above Möbius-invariant metric we can again use the route from Section 9.3.

Lecture 9 Invariant Metric and Geodesics

9.1 Metrics, Curves’ Lengths and Extrema

9.2 Invariant Metric

9.3 Geodesics: Additivity of Metric

9.4 Geometric Invariants

9.5 Invariant Metric and Cross-Ratio