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Conformal Geodesics Cannot Spiral

We show that conformal geodesics on a Riemannian manifold cannot spiral: there does not exist a conformal geodesic which becomes trapped in every neighbourhood of a point.


Introduction
A pseudo-Riemannian manifold, (M, g) defines a distinguished set of curves which are metric geodesics of g. These curves are solutions to a set of 2nd order ODEs and are uniquely specified locally by an initial point and an initial tangent direction. However, with the exception of null geodesics in the Lorentzian case, there is in general no relation between the metric geodesics of two conformally related metrics. By analogy with metric geodesics, conformal geodesics can be thought of as a distinguished set of curves defined on a conformal manifold (M, [g]). These curves are solutions to a system of conformally invariant 3rd order ODEs and are uniquely specified locally by an initial position, unit tangent direction and perpendicular acceleration [11,1,2,10]. Although metric geodesics have been studied extensively, relatively little is known about conformal geodesics.
Conformal geodesics were systematically introduced into general relativity by Friedrich and Schmidt [6]. Their motivation was to construct good coordinate systems for the local study of conformal boundaries, as introduced in Penrose's process of conformal compactification. Normal or Gaussian coordinates based on timelike geodesics cannot be expected to behave well near such conformal boundaries -recall for example how, in compactified Minkowski space, timelike geodesics do not pass through null infinity but instead all focus at timelike infinity. It was shown in [6] (following an earlier suggestion in [9]) that one could instead define conformal normal or conformal Gaussian coordinates constructed from conformal geodesics. These would provide good coordinates near the conformal boundaries of asymptotically Minkowskian, asymptotically de Sitter or anti-de Sitter spacetimes.
An unresolved issue stemming from this earlier work, which was also known as an unsolved problem in Riemannian-signature conformal geometry, was the question of spiralling for conformal geodesics. A curve can be said to spiral at a point p * if it enters and remains in every neighbourhood of p * but does not pass through p * itself. It is a classical result, based on the existence of geodesically-convex neighbourhoods [5], that metric geodesics cannot spiral, although until now this has not been shown for conformal geodesics. It is a question of interest both abstractly and because spiralling raises the possibility of a new kind of coordinate singularity to guard against.
In [10], spiralling was ruled out in some special cases using 1st integrals, and integrability (see also [3] for integrable, non-spiraling examples on four-manifolds). For example, it was shown that conformal geodesics defined on a Riemannian Einstein manifold cannot spiral. The conformal geodesic equations were also integrated for some example geometries and it was observed that the solutions did not spiral.
In this paper we will prove a general no-spiralling theorem for conformal geodesics (Theorem 2.3). We begin by considering the analogous result for metric geodesics. The standard proof that metric geodesics cannot spiral relies on the existence of a geodesically convex neighbourhood at each point [5,Proposition 4.2]. Such a proof will not work for conformal geodesics since, despite their variational origins [4], these are not local length minimisers. With this in mind, we begin by constructing a proof of the no-spiralling theorem for metric geodesics which does not rely on length minimisation arguments. This will then form the basis of our proof in the conformal geodesic case.
conformally invariant third order differential equations [10]: where u is the unit tangent vector to the curve (i.e g(u, u) = 1), a := ∇ u u is the perpendicular acceleration (i.e g(u, a) = 0) and |a| := g(a, a). The Schouten tensor L : in terms of the Ricci tensor Ric, the Ricci scalar R and the dimension of the manifold d. We denote by L # : T p M → T p M the corresponding endomorphism defined by g(L # u, v) = L(u, v) for all vector fields u, v.
Changing the metric toĝ = Ω 2 g, where Ω : M → R results in changes to the Schouten tensor, the Levi-Civita connection, unit tangent vector and the acceleration where Υ ≡ Ω −1 dΩ, and Υ is a vector field defined by Υ(w) = g(Υ , w). It is now a matter of explicit calculation to verify that the conformal geodesic equations (2.1) are conformally invariant.
In the rest of the paper we shall fix a metric g in the conformal class, and consider (2.1) for this choice of metric. We will denote the set of unit vectors in T p M by S(T p M ) and S (T M ) will denote the corresponding unit tangent bundle over M .
Picard's theorem applied to equation (2.1) shows that, locally, a conformal geodesic is uniquely specified by an initial point p ∈ M , an initial unit tangent, u 0 , at p and an initial acceleration, a 0 , at p which is perpendicular to u 0 [10, Theorem 1.1]. We will denote the conformal geodesic with these initial conditions and arc length parameter t by γ p,u0,a0 (t).
Definition 2.1. A curve, γ, with arc length parameter t, spirals towards a point p * ∈ M if for any neighbourhood N containing p * , there exists some T such that γ(t) ∈ N for all t > T .
We demand that the conformal geodesic has a well defined unit tangent vector and thus exclude the trivial case where it consists only of a single point.
Throughout this paper we will refer to balls in (M, g) and in various tangent spaces. We make the following definitions where d : M × M → R ≥0 denotes the Riemannian distance function induced by the metric g.
We will also refer to balls in the tangent space T u0 (T p M ), where u 0 ∈ S(T p M ). We identify T u0 (T p M ) with T p M and these balls are defined using the metric g, which in turn defines a notion of length on the vector space T p M .

Outline of Proof for Metric Geodesics
Theorem 2.2. Let (M, g) be a Riemannian manifold. Then geodesics on (M, g) cannot spiral.
Sketch of proof (without using length minimisation arguments): • Step 1: Define the exponential map and geodesic ball at p ∈ M and establish that all metric geodesics through p reach the boundary of this geodesic ball.
The exponential map, exp p at a point p ∈ M [8, Definition 1.10] is a diffeomorphism from some neighbourhood of U ⊂ T p M containing 0 to a neighbourhood V ⊂ M containing p. A geodesic ball is any open ball centred on p and contained in V . The exponential map at p allows us to construct a co-ordinate system on any geodesic ball. Geodesic segments foliate the geodesic ball with the point p removed. For the purposes of this proof, the important property of the geodesic ball is that all metric geodesics through p must reach its boundary. • Step 2: Show that for any compact U ⊂ M there exists > 0 such that for all p ∈ U , B(p, ) is a geodesic ball.
We define a function on M which maps each point p ∈ M to the supremum of the set {r ∈ R ≥0 : B(p, r) is a geodesic ball} (capped at 1 to avoid any complications involving infinite sizes). Since the exponential map defined at any p ∈ M is a local bijection, this function is always strictly positive. It can also be shown that this function is continuous. The result follows from the fact that this continuous function restricted to the compact set U is bounded from below by some > 0.
• Step 3: Use the fact that metric geodesics intersect the boundary of the geodesic ball to deduce that they cannot spiral.
Suppose the metric geodesic γ spirals towards some p * ∈ M . Then for any R < d(p * , γ(0)) there is a unique T > 0 such that d(p * , γ(T )) = R and d(p * , γ(t)) < R for all t > T . Denote the point γ(T ) by p. This is the point at which γ enters B(p * , R) for the final time.
Let U ⊂ M containing p * be compact. Choose R = /2 (where is as in Step 2) and find the corresponding p (reducing if necessary so that R < d(p * , γ(0)). By reducing further if necessary, we can assume that p ∈ U . Then, using the results of Step 2, B(p, ) is a geodesic ball. From Step 1 we know that γ must leave this geodesic ball at some later parameter value. In doing so, it must once again reach a distance of at least R = /2 from p * . This is a contradiction since we assumed that d(p * , γ(t)) < R for all t > T . This argument is illustrated in Figure 1.

Outline of Proof for Conformal Geodesics
The exponential map at p ∈ M was useful because it allowed us to identify a subset of M which was foliated by segments of metric geodesics through p. Inspired by this, we will define an exponential Figure 1: We find T ∈ R and > 0 such that there is a geodesic ball of radius centred at p := γ(T ), where d(p * , γ(T )) = /2 and d(p * , γ(t)) < /2 for all t > T . However, we have also shown that γ must leave this geodesic ball and hence reach a distance of /2 from p * at some t > T . This is our contradiction. We show geodesic balls as circles for ease of illustration.
map adapted to conformal manifolds which allows us to see how conformal geodesics foliate some subset of M .
Despite some additional complexities in the conformal geodesic case, we are able to prove a no-spiralling theorem by following three steps analogous to those used in the proof of Theorem 2.2. These three steps correspond to the three main sections of this paper. Outline of Proof (see Section 5 for the complete proof using results derived in Sections 3 and 4): • Step 1: Define an exponential map adapted to conformal geodesics and identify a domain and range on which it is a homeomorphism.
In Section 3, we define a conformal geodesic analogue of the exponential map at some (p, u 0 ) ∈ S(T M ). Unlike for metric geodesics, there is no subset of the domain of this exponential map on which it is a homeomorphism onto some ball centred at p. Instead, in Section 3.3 we find that it is a homeomorphism onto some set which we call a heart p. We say that this heart has direction u 0 . Crucially, in Section 3.4 we are able to show that conformal geodesics re-intersect the boundary of this heart. This is analogous to the fact that metric geodesics through some point re-intersect the boundary of a geodesic ball centred at that point.

Result of
Step 1 (Theorem 3.7): For any (p, u 0 ) ∈ S(T M ), any conformal geodesic with unit tangent u 0 at p re-intersects the boundary of a heart at p with direction u 0 .
• Step 2: Define a notion of size for a heart and show that given any compact U ⊂ M , there exists > 0 such that there is a heart at p with direction u 0 which has size at least , for any p ∈ U and any u 0 ∈ S(T p M ).
In the metric geodesic case we considered geodesic balls in M since these had an obvious size associated to them. In the conformal geodesic setting it is simpler to consider certain half balls in T u0 (T p M ) which are mapped to hearts in M . The size of a heart is defined by considering the radii of balls in M which have u 0 tangent to their boundary and are contained inside this heart. Note that this size has units of length, regardless of the dimension of the conformal manifold. We show that as we vary p and u 0 (as well as the radius of the domain ball 1 ) in T u0 (T p M ) we are able to find a lower bound for this size which is strictly positive and varies continuously with p and u 0 (provided the Schouten tensor is of class C 2 ). The result follows from the fact that a continuous function on a compact set is bounded and attains its bound.

Result of
Step 2 (Theorem 4.2): Let (M, [g]) be a conformal manifold such that the Schouten tensor is of class C 2 and let U be a compact subset of S(T M ). Then there exists some > 0 such that at any (p, u 0 ) ∈ S(T U ) there is a heart H p,u0 at p with direction u 0 which has size at least .
• Step 3: Using the fact that a conformal geodesic with unit tangent u 0 at p must re-intersect the boundary of H p,u0 , deduce that conformal geodesics cannot spiral.
Suppose γ is a conformal geodesic which spirals towards some p * ∈ M . We consider a metric geodesic ball centred on p * (i.e. a set B(p * , r) for some r) which is entered by γ for the final time at some point p where it has unit tangent u 0 (i.e. γ enters the ball at p and never re-intersects its boundary). Using steps 1 and 2, we show that it is possible to choose this ball (and the corresponding p and u 0 ) such that it is contained inside some heart H p,u0 with position p and direction u 0 (see Definition 3.6). By Theorems 3.7 and 4.2, γ must re-intersect the boundary of this heart, so we deduce that it must also re-intersect the boundary of the ball. This is our contradiction.

The Exponential Map for Conformal Geodesics
In this section we will complete Step 1 of the proof of Theorem 2.3 as outlined in Section 2.1.
Step 1: Define an exponential map adapted to conformal geodesics and identify a domain and range on which it is a homeomorphism.
For any (p, u 0 ) ∈ S (T M ), we define the following analogue of the exponential map appropriate to conformal geodesics (see [8,Definition 1.10] for a definition of the exponential map for metric geodesics): This map is continuous as a function of A, as we will see in the next section.

The Directional Derivative of The Exponential Map
The exponential map for metric geodesics is usually analysed by considering its directional derivatives. We will do the same thing for the conformal geodesic exponential map in Definition 3.1. Calculating these directional derivatives at A = 0 will allow us to understand this map in neighbourhoods of p.
The directional derivative at 0 is To calculate this, we use a power series expansion: where ∇ (n) u denotes the directional derivative ∇ u applied n times and we recall that a = ∇ u u (so in particular we set ∇ u u| t=0 = a 0 ). This expression (and subsequent similar expressions) should be understood in terms of co-ordinates.
Using repeated applications of the conformal geodesic equation (2.1), we can express the power series (3.6) in terms of u 0 and a 0 (as well as the Schouten tensor and its derivatives evaluated at p). From equation (2.1), if the initial acceleration is a 0 = A ⊥ / λ|A| 2 , then at t = 0 we have where u 0 denotes the initial unit tangent velocity and by O(1) we mean terms which are bounded in the limit λ → 0. From this we see that for a conformal geodesic with initial data p, u 0 , A ⊥ / λ|A| 2 , we have as λ → 0. In particular all terms involving the Schouten tensor L and its derivatives are O(λ 2−n ) (i.e. they are sub-leading in the limit λ → 0). To calculate the directional derivative (3.5) we must evaluate the power series (3.6) at t = 2πλ|A| = O(λ). We see that the terms in this power series which are first order in λ are exactly those that arise in Euclidean space, where L ≡ 0. For the Euclidean metric on R n , conformal geodesics with initial data (p, u 0 , a 0 ) are circles with equations of the form The directional derivative of the exponential map at 0 is therefore Note that this expression is finite in the limit A ⊥ → 0.
It is clear from the definition of the exponential map that if A and A are related by a reflection in a plane perpendicular to u 0 (i.e. A ⊥ = A ⊥ and |A| = |A |), then exp p,u0 (A) = exp p,u0 (A ). Since our aim is to find a region on which the exponential map is a homeomorphism, we restrict our attention to the half space g(A, u 0 ) ≥ 0. Based on equation (3.10), we define the following quantities related (3.11) We have assumed that A ⊥ = 0 and A = 0 since this will simplify expressions later on, although these cases can be obtained from the results below by taking the limits A ⊥ → 0 and A → 0 respectively.
We see that a vector A in the half space g(u 0 , A) ≥ 0 is uniquely defined by θ,â 0 and |A|. The vectorâ 0 defines a "quarter space" in T p M by g(A,â 0 ) ≥ 0 and g(A, u 0 ) ≥ 0. The variable θ is the angle between A and the vector 2 u 0 .
In terms of |A|, θ andâ 0 , we can re-write the directional derivative (3.10) as D(exp p,u0 ) 0 : A → u 0 |A| sin θ sin (2π sin θ) +â 0 |A| sin θ (1 − cos (2π sin θ)) . (3.12) We therefore have the following leading order expression for the exponential map If we holdâ 0 and sin θ/|A| fixed while varying |A|, we see from Definition 3.1 that the image of this curve under the exponential map describes a conformal geodesic parameterised by |A|. Suppose we extend u 0 ,â 0 to obtain an orthonormal basis of T p M . Using the metric geodesic exponential map, this basis corresponds to a set of geodesic normal co-ordinates for a neighbourhood of p ∈ M . This co-ordinate expression for the conformal geodesic is the same as the co-ordinate expression for a circle in R n to first order in |A|, with the O(|A| 2 ) terms vanishing. Recall that conformal geodesics in Euclidean space are exactly circles (all higher order terms in |A| vanish). By analogy, if we do a power series expansion of metric geodesics in these same co-ordinates we also find that the 2nd order term in the expansion parameter vanishes. Recall that metric geodesics in Euclidean space are exactly straight lines (i.e. there exist co-ordinates in which all non-linear terms in the expansion parameter vanish).

The Image of a Ray Under the Exponential Map
In this section we will study the image of a ray in the half space g(u 0 , A) ≥ 0 originating from A = 0. Such a ray is defined by fixing θ andâ 0 . As in Section 3.1,â 0 defines a quarter space in T p M via the conditions g(u 0 , A) ≥ 0, g(â 0 , A) ≥ 0 and θ denotes the angle between A and u 0 .
. Under the exponential map at (p, u 0 ), a ray from A = 0 defined byâ 0 and θ = θ 0 is mapped to a curve in M whose tangent at p lies in the half space of span{u 0 ,â 0 } defined by g(u 0 ,â 0 ) ≥ 0 and makes an angle of θ = π sin θ 0 with u 0 (see Figure 2).

Proof:
The directional derivative at 0 in the direction tangent to a particular ray is given by equation (3.12) and is non-zero provided A = 0 and θ = π/2 (i.e. A ⊥ = A). This directional derivative gives the tangent direction at p of the curve which is the image of this ray under the exponential map. We see that this image curve lies in the same quarter space as A and that the angle, θ, between the directional derivative and the vector u 0 satisfies tan θ = 1 − cos(2π sin θ 0 ) sin(2π sin θ 0 ) = tan(π sin θ 0 ) =⇒ θ = π sin θ 0 . (3.14)

The Exponential Map is a Homeomorphism onto a Heart
Lemma 3.2 tells us that the exponential map takes distinct rays through the origin in T p M to curves with distinct tangent directions at p. We will show that for rays confined to a wedge in T p M , we can find a lower bound on the arc length parameter (equivalently a lower bound on |A| in Definition 3.4) at which their image curves can re-intersect. This tells us that the exponential map is a homeomorphism on this wedge. We will then consider a union of such wedges. We find that as we increase the opening angle of the wedge towards π, the lower bound on the arc length parameter we obtain decreases to 0. We conclude that the exponential map is a homeomorphism on some region in the half space g(A, u 0 ) ≥ 0 with cross-sections like the one shown on the right of Figure 3.     Proof: As described above, a ray in the half space g(u 0 , A) ≥ 0 is specified by fixingâ 0 and θ. The variable |A| then acts a a parameter along this ray. There are therefore two types of variation we can make to obtain neighbouring rays. Firstly, we can vary θ. To obtain the deviation vector connecting the image of such infinitesimally separated rays, we differentiate equation (3.13) with respect to θ, keepingâ 0 constant. The other variation we can consider is varyingâ 0 (while keepinĝ a 0 perpendicular to u 0 ). In this case, we obtain a deviation vector which points in some other direction, perpendicular to both u 0 andâ 0 , which we denoteâ 0 (defined to have unit length). Considering both of these variations together, we obtain a deviation vector: Note that this deviation vector is finite at θ = 0. We also consider the tangent vector, T , to the image of a ray in T p M . This is obtained by differentiating equation (3.13) with respect to |A| while holding θ constant: What we are actually interested in is the component of S which is perpendicular to T . We therefore define whereT denotes the unit vector T /|T |.
Since we are interested in the values of |A| at which this might vanish, we differentiate with respect to |A|, holding θ andâ 0 constant: This function is shown in Figure 5. If we restrict to a wedge in T p M defined by θ ∈ [θ 0 , θ 1 ] for some 0 < θ 0 < θ 1 < π/2, then ∂S ⊥ ∂|A| is bounded away from 0 by some constant c 0 > 0 (which depends on θ 0 and θ 1 ). Furthermore, if we consider only points in the wedge with |A| ≤ r (i.e. a compact subset of the wedge which we will refer to as a wedge of radius r, then ∂ 2 S ⊥ ∂|A| 2 is bounded above by some constant c 1 > 0 (since all terms in the power series expansion (3.17) will be bounded on this compact set). We therefore have where in the last line we have used the fact that |S ⊥ || |A|=0 = 0.
From this we see that the exponential map is a homeomorphism on the set of points in T p M with θ ∈ [θ 0 , θ 1 ] and |A| ≤ c 0 /c 1 .
We can repeat this argument for any choice of θ 0 and θ 1 with 0 < θ 0 < θ 1 < π/2. However, if we let θ 0 → 0 and θ 1 → π/2, it is clear from Figure 5 that c 0 → 0. Hence the radius of the wedge on which we can guarantee that the exponential map is a homeomorphism shrinks to 0. This shows that the exponential map is a homeomorphism on each of a series of wedges with radii shrinking to 0 as their opening angles increase to π. This is illustrated in Figure 3.
By taking the union of these wedges, we conclude that the exponential map is a homeomorphism on some set with cross-sections which have boundary perpendicular to u 0 at p (see Figure 3). Such a set contains an open ball in T u0 (T p M ) which has u 0 inward pointing and perpendicular to its boundary. As the radius of a wedge shrinks to 0, so too must the size of the smallest ball which contains its image under the exponential map. Furthermore, by Lemma 3.2 we see that as the opening angle of this wedge increases to π, the opening angle of the image set at p must increase to 2π (consider two rays defined by ±â 0 and let θ 0 → π). In particular, the images of these two rays are both tangent to −u 0 at p. Consequently, the image of a ball with u 0 inward pointing and perpendicular to its boundary is a set with cross sections which feature a cusp at p (shown on the right of Figure 4). We refer to such a set as a heart due to the shape of these cross sections.
Note that if we choose |A| = |A ⊥ | = , then the vector A will be mapped to a point on a conformal geodesic which has perpendicular acceleration of size 1/ at p. This can be made arbitrarily large by choosing to be arbitrarily small. As a result, given any r > 0, the image set exp p,u0 (C(u 0 , r)) will always contain at least some portion of every conformal geodesic with unit tangent vector u 0 at p, even those with arbitrarily large perpendicular acceleration at this point. This is crucial for the lemma below and for our proof of the no-spiralling theorem (Theorem 2.3) in Section 5.
Example (the Euclidean metric on R n ): In the case where g is the Euclidean metric on R n , the O(|A| 3 ) terms in equation (3.13) are identically 0 and conformal geodesics are circles. The images of distinct rays from 0 in the half space g(u 0 , A) > 0 are also rays. These do not re-intersect, so the exponential map is a homeomorphism (in fact a diffeomorphism) from this half space onto R n with the ray from 0 in the direction of −u 0 removed. This is essentially the familiar result that we can foliate R n by circles tangent to u 0 at p (shown for n = 2 in Figure 6). Since we have insisted that the conformal geodesic has initial direction u 0 (rather than −u 0 ) we cannot reach points on the −u 0 axis. These points correspond to the circle through infinity with initial direction u 0 at p.
If the Schouten tensor is not indentically zero then the higher order terms in equation (3.13) become important as |A| increases. These terms may cause the images of the rays to intersect. This would reduce the size of the domain on which the exponential map is a homeomorphism.

Conformal geodesics re-intersect the heart boundary
Given any (p, u 0 ) ∈ S (T M ), we would like to consider the largest open ball C(u 0 , r) (see Definition 3.3) on which exp p,u0 is a homeomorphism. It is of course possible this map is a homeomorphism on C(u 0 , r) for all r > 0, as is the case in Euclidean space (see Section 3.3). Our proof of the no-spiralling theorem in Section 5 will be simpler if we work with a finite sized domain with boundary. With this in mind, we make the following definition The factor of 1/2 is included since it will be convenient in the proof of Theorem 4.2 that the exponential map is a homeomorphism onto the heart and its boundary.
Definition 3.6. The heart which is the image of the set C(u 0 , r (p,u0) ) under the exponential map at (p, u 0 ) is denoted H p,u0 := exp p,u0 (C(u 0 , r (p,u0) )). (3.22) For the proof of the no-spiralling theorem (Theorem 2.3), we will require the following property of the heart H p,u0 .
Theorem 3.7. For any (p, u 0 ) ∈ S(T M ), any conformal geodesic with unit tangent u 0 at p re-intersects the boundary of the heart H p,u0 .
Proof: As discussed in Section 3.1, a conformal geodesic is the image under the exponential map of a curve in T p M defined by holdingâ 0 and sin θ/|A| constant while varying |A|. Therefore, a conformal geodesic intersects the boundary of H p,u0 when this curve reaches the boundary of C(u 0 , r (p,u0) ) ⊂ T p M .
In fact, the curve in T p M defined by holdingâ 0 and sin θ/|A| constant reaches the boundary of C(u 0 , r (p,u0) ) at some |A| ≤ r (p,u0) . Hence from the definition of the exponential map (Definition 3.1) we see that the arc length parameter distance travelled along the conformal geodesic before it reaches the boundary of H p,u0 is at most 2πr (p,u0) .

Size of the Heart
The aim of this section is to follow Step 2 of the proof of Theorem 2.3 as outlined in Section 2.1.
Step 2: Define a notion of size for a heart and show that given any compact U ⊂ M , there exists > 0 such that there is a heart at p with direction u 0 which has size at least , for any p ∈ U and any u 0 ∈ S(T p M ). Consider the heart H p,u0 and let k(p, u 0 , r) denote the set of all closed balls in M of radius r with p on their boundary and u 0 pointing into or tangent to the ball. If r is sufficiently small, then all balls in k(p, u 0 , r) will be contained inside the closure of H p,u0 , which we denoteH p,u0 (see Figure 7). To prove this theorem, we begin by considering the following ODE with an initial condition y (t) = f (t, y) and y(0) = y 0 . Corollary 4.4. Suppose the Christoffel symbols, Γ β γδ and the Schouten tensor, L, are both C 2 . Then the heart boundary is a C 2 surface away from the cusp.
Proof: We will parameterise conformal geodesics x α (t) by their arc length, t. The conformal geodesic equation can then be written as three separate 1st order ODEs on some co-ordinate patch (4.25) Since the Christoffel symbols Γ β γδ and Schouten tensor are both C 2 , Theorem 4.3 implies that the solution x α = η α (t, x 0 , u 0 , a 0 ) is also C 2 . This tells us that, away from A = 0, the exponential map considered as a function of (p, u 0 , A) is C 2 . We conclude that the heart boundary is C 2 away from the cusp.
Next we define the following function.  Proof: Let U ⊂ S(T M ) be compact and let (p, u 0 ) ∈ U . This specifies a value r (p,u0) such that the exponential map at (p, u 0 ) is a homeomorphism from the open ball C(u 0 , r (p,u0) ) to the heart H p,u0 . Next we choose a unit vectorâ 0 which is perpendicular to u 0 . By restricting to elements of T p M in span{u 0 ,â 0 }, we obtain a curve through p on the boundary of the ball C(u 0 , r (p,u0) ) ⊂ T u0 (T p M ) which we call a circle of radius r (p,u0) . This circle is mapped by the exponential map to a curve λ p,u0,â0 on the heart boundary. Finally, we choose some value x ∈ (−π/2, π/2). This corresponds uniquely to a point on this circle (by considering the angle between A and u 0 ). The exponential map takes this point to a point on the curve λ p,u0,â0 . Note that by choosing x ∈ (−π/2, π/2) we do not include the problem point at 0 which is mapped to the cusp.
By Corollary 4.4, λ p,u0,â0 is an endless C 2 curve. This allows us to consider its curvature. For any x ∈ (−π/2, π/2) we define the following real-valued function There is a subtlety at the point p, where the boundary of the heart contains a cusp and directional derivatives have discontinuities. However, if we approach p along the curve λ p,u0,â0 in two different directions then the function F tends to a finite limit (note that these two limits may not be equal).
If we join the two ends of λ p,u0,â0 at p, then these two limits correspond to the two values we get for the magnitude of the curvature of λ p,u0,â0 at p if we use one-sided derivatives in either direction. Figure 8: If the every sectional curvature at every point on the heart boundary is bounded above by > 0, then any ball of radius 1/ with u 0 tangent to its boundary at p is contained inside the heart.
In short, given any (p, u 0 ) ∈ U and any circle of radius r (p,u0) in T u0 (T p M ) with u 0 perpendicular to its boundary, the magnitude of the sectional curvature at a point on the image of this circle, taken in the direction of the circle's image, is bounded by 1/ > 0, for some > 0, where we understand that at p itself we consider two different sectional curvatures corresponding to the different directions along the image of the circle.
As a result, for any (p, u 0 ) ∈ U , the heart H p,u0 must contain every open ball in M of radius which has u 0 tangent to its boundary at p, i.e. R 1 (p, u 0 ) ≥ (see Figure 8).
Next we consider balls in M with u 0 pointing inwards on the boundary. We make the following definitions. Here exp p denotes the metric geodesic map at p (we can restrict ourselves to sufficiently small r so that this map is invertible on B(p, r) for all (p, u 0 ) ∈ W ). The set we have defined consists of points in the half ball with boundary perpendicular to u 0 at p which have non-negative u 0 co-ordinate, where these co-ordinates are defined using the metric geodesic map applied to an orthonormal basis of T p M (see Figure 9).   Proof: Let U ⊂ S(T M ) be compact. From Corollary 4.4, exp p,u0 (A) regarded as a function of (p, u 0 , A) is continuous (in fact C 2 ) away from the cusp. We also saw in Section 3.5 that the boundary of H p,u0 is continuous (including at p). We conculde that the boundary of H p,u0 varies continuously with (p, u 0 ) and hence R 2 (p, u 0 ) is also continuous. The result follows since R 2 (p, u 0 ) is strictly positive and a continuous function on a compact set is bounded and attains its bounds.
Combining these two lemmas, we are able to prove Theorem 4.2.

No-Spiralling Theorem
The aim of this section is to follow Step 3 of the proof of Theorem 2.3 as outlined in Section 2.1.
Step 3: Using the fact that a conformal geodesic with unit tangent u 0 at p must re-intersect the boundary of H p,u0 , deduce that conformal geodesics cannot spiral. Proof: Suppose there is a conformal geodesic, γ, which spirals towards some point p * ∈ M . We will show that there exists r > 0 such that B(p * , r) ⊂ H p,u0 , where p and u 0 are the point and unit tangent at which γ entersB(p * , r) for the final time. This would then contradict Theorem 3.7 since if γ does not leave B(p * , r) then it cannot re-intersect the boundary of H p,u0 . This is illustrated in Figure 11. So for any (p, u 0 ) ∈ S (T M ) with d(p, p * ) ≤ 1, any closed ball in M of radius at most R W and u 0 inward pointing or tangent to its boundary at p is contained insideH p,u0 .
Let p denote the point at which γ entersB(p * , R W ) for the final time and let u 0 denote its unit tangent at this point (so u 0 is either inward pointing or tangent to the boundary ofB(p * , R W ) at p). By construction we then haveB(p * , R W ) ⊂ H p,u0 . But since γ remains trapped in B(p * , R W ), it cannot re-intersect the boundary of H p,u0 . This contradicts Theorem 3.7. Figure 11: If γ enters B(p * , R W ) ⊂ H p,u0 , then in order to reach the boundary of H p,u0 it must first leave B(p * , R W ).

Summary
We have shown that conformal geodesics defined on a Riemannian conformal manifold cannot spiral. This proof was inspired by a proof of the no-spiralling theorem for metric geodesics which did not rely on length minimisation arguments, instead focusing on properties of the exponential map. However, in the conformal geodesic version there were some additional complications. The most striking of these was that the exponential map we defined for conformal geodesics was not a homeomorphism onto any ball centred at p ∈ M . Instead we found that given any (p, u 0 ) ∈ S (T M ), conformal geodesics with unit velocity u 0 at p foliate some set with cross-sections which resemble a heart.
In this paper we have only considered Riemannian conformal manifolds. In the Lorentzian case, our proof fails from the very beginning. In particular, the exponential map (Definition (3.1)) is not well defined since it is now possible to have |A| = 0 for A = 0.