The Co-Points of Rays are Cut Points of Upper Level Sets for Busemann Functions

We show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, and other properties follow immediately from the known results about the cut locus. We point out that some of our findings, in special the relation of co-point set to the upper lever sets, are new even for Riemannian manifolds.


Introduction
Roughly speaking, Busemann function is a function that measures the distance to a point at infinity on a complete non-compact Riemannian or Finsler manifold. Originally introduced by H. Busemann for constructing a theory of parallels for straight lines (see [Bu1], [Sh], [In1], [In2]), the function playes a fundamental role in the study of complete non-compact Riemannian or Finsler manifolds ( [Oh], [Sh], [SST], etc).
In the present paper we study the differentiability of the Busemann function in terms of corays and co-points to a ray in the general case of a forward complete non-compact Finsler manifold. We show that the notions of geodesic segments to a closed subset and the cut locus of such sets can be extremely useful in the study of corays and co-points to a ray, that is points where Busemann function is not differentiable.
The originality of our research is two folded. Firstly, the detailed study of Busemann functions, corays and co-points on Finsler manifolds is new. Secondly, in the special case of Riemannian manifolds, our Main Theorems 1.1 and 1.2, first statement, are new and they lead to new elementary proofs of other results already known.
A Finslerian unit speed globally minimizing geodesic γ : [0, ∞) → M is called a (forward) ray. A ray γ is called maximal if it is not a proper sub-ray of another ray, i.e.
1. α is a subarc of a coray of γ.
From here it naturally follows the relation between the co-points to a forward ray and the cut points of a level set of Busemann function. 2. The Busemann function b γ is differentiable at a point x of M if and only if x admits a unique coray σ to γ emanating from x = σ(0). In this case ∇b γ (x) =σ(0).
These two Main Theorems reduce the study of corays and co-points to the study of N-segments and cut points of a closed subset of (M, F ), respectively, making in this way possible to apply our previous results from [TS].
Seen in this light, the structure theorem for the co-points set on a Finsler surface, namely that is a local tree, it becomes trivial. It is also clear that the topology of (C γ , δ), with the induced metric, coincides with the topology of the Finsler surface, as well as that (C γ , δ) is forward complete (see Theorem 2.11). Other results are also straighforward from [TS] (see Theorem 2.12).
In the final Section we construct examples of Finsler manifolds whose co-points set C γ can be explicitly determined (Examples 4.1,4.2). From here it follows that there exists forward complete Finsler metrics (M, F ) containing rays γ whose set of co-points C γ is not closed.
Acknowledgments. The author is grateful to Prof. M. Tanaka for bringing this topic into his attention and for many illuminating discussions. Also I thank to N. Boonnam for reading an early version of the paper.

Busemann functions
Let (M, F ) be a forward complete non-compact Finsler manifold (see [BCS], [S] for details on the completeness of Finsler manifolds). In Riemannian geometry, the forward and backward completeness are equivalent, hence the words "forward" and "backward" are superfluous, but in Finsler geometry these are not equivalent anymore.
Definition 2.1 If γ : [0, ∞) → M is a ray in a forward complete non-compact Finsler manifold (M, F ), then the function is called the Busemann function with respect to γ, where d is the Finsler distance function.
The Busemann function for Finsler manifolds were introduced and partially studied by Egloff [Eg] and more recently by [Oh].
It follows that a point x of M is an element of b −1 γ (a, ∞), for some real number a, if and only if t − d(x, γ(t)) > a for some t > 0, and hence we get x, γ(t + a)) < t} denotes the backward open ball centered at γ(t + a) of radius t. In particular b −1 γ (a, ∞) is arcwise connected for each a ≥ 0. Triangle inequality shows for any two points x, y ∈ M.
The differentiability of Busemann function is fundamental for the study of corays. Some results are already known. Let us denote by ∇f (x) the Finslerian gradient of a smooth function f : M → R (see [Oh], [S]).
Theorem 2.5 [Oh] Let γ be a forward ray in a non-compact forward complete Finsler manifold (M, F ).
1. For any x ∈ M, there exists at least an coray σ of γ such that σ(0) = x.
3. If b γ is differentiable at a point x ∈ M, then σ(s) := exp x (s∇b γ (x)) is the unique coray to γ emanating from x, where ∇b γ (x) is the Finslerian gradient of b γ at x.
We point out that the converse of the statement 2 in Theorem 2.5 is also true, but we not need it here.
For any closed subset N of M, we have defined N-segments in Introduction. From now on, any N a -segment will mean forward N a -segment, where N a := b −1 γ [a, ∞).
2 ⇒ 3. Choose any s ∈ [0, a] and any On the other hand, hypothesis (1.1) implies From relations (2.3) and (2.4) it results d(α(s), α(a)) = d(α(s), N b ) for any s ∈ [0, a], and since the point x is arbitrarily chosen from N b we obtain that α is an N b -segment.
2. Follows easily from Theorem A in [TS], Corollary 2.7 and Theorem 1.1. ✷ Proof. Choose any point σ(t 0 ), t 0 > 0. By Theorem 1.1, the subray σ| [t 0 ,∞) is a unique coray of γ emanating from σ(t 0 ). Thus, by Theorem 1.2, statement 2 it results that b γ is differentiable at σ(t 0 ). ✷ Let us denote by C γ the set of co-points of the ray γ, that is the origin points of the maximal corays to γ. By Proposition 2.6 in [TS] and Theorem 1.1 we obtain, Corollary 2.9 Let (M, F ) be a forward complete Finsler manifold, γ a forward ray in M and C γ the set of co-points of γ.
Then the subset In the two dimensional case, the structure equations of the cut locus from [TS] can be easily extendend. We recall that an injective continuous map from the open interval (0, 1) (or closed interval [0, 1]) of R and from a circle S 1 into M is called a Jordan arc and a Jordan curve, respectively.
A topological set T is called a tree if any two points in T can be joined by a unique Jordan arc in T . Likely, a topological set C is called a local tree if for every point x ∈ C and for any neighborhood U of x, there exists a neighborhood V ⊂ U of x such that C is a tree. A point of a local tree C is called an endpoint of the local tree if there exists a unique sector at x.
A continuous curve c : is finite. By Theorem 1.2 and Theorem B in [TS] we obtain (compare with [L]) Theorem 2.11 Let γ be a ray in a forward complete 2-dimensional Finsler manifold (M, F ). Then the set C γ of co-points of γ satisfies the following three properties.
1. The set C γ is a local tree and any two copoints on the same connected component of C γ can be joined by a rectifiable curve in C γ .
2. The topology of C γ induced from the intrinsic metric δ coincides with the induced topology of C γ from (M, F ).
3. The space C γ with the intrinsic metric δ is forward complete.
Indeed, by the first statement, any two copoints y 1 , y 2 ∈ C γ can be joined by a rectifiable arc in C γ if y 1 and y 2 are in the same connected component. Therefore, the intrinsic metric δ on C γ defined as: inf{l(c)| c is a rectifiable arc in C γ joining y 1 and y 2 }, if y 1 , y 2 ∈ C γ are in the same connected component, +∞, otherwise is well defined. By Theorem 1.2 and Theorem C in [TS] we have Theorem 2.12 Let γ be a ray in a forward complete 2-dimensional Finsler manifold (M, F ). Then there exists a set E ⊂ [0, ∞) of measure zero with the following properties: 1. For each t ∈ (0, ∞) \ E, the set b −1 γ (t) consists of locally finitely many mutually disjoint arcs. In particular, if b −1 γ (a), is compact for some a > t, then b −1 γ (t) consists of finitely many mutually disjoint circles.

Implications of the differentiability of b γ
Here are some results that follows from the previous section (compare with [In1]). In [In1] it is proved for G-spaces that if C γ is compact, then b γ is an exhaustion function. We will give a more general result.
Theorem 3.1 Let (M, F ) be a forward complete non-compact Finsler manifold and γ a ray in M.
If there exists a divergent numerical sequence {c i } such that then the Busemann function b γ is an exhaustion function, i.e. for any a ∈ R, the set b −1 ((−∞, a]) is compact.
Proof. For each fixed i and c i , we define the set γ (c i ) | q belongs to some coray to γ emanating from a point in C γ }.
We claim that for any fixed i, we have S i = b −1 γ (c i ). We will prove this claim by showing that S i is both closed and open in b −1 γ (c i ). Indeed, firstly, we show that S i is closed. If we consider a convergent points sequence {q j } ⊂ S i , then we will show that q := lim j→∞ q j belongs to S i . If we denote by {σ j } the corresponding corays to γ emanating from the initial points {x j } ⊂ C γ and passing through {q j }, respectively, then one can see that actually {x j } ⊂ C γ ∩ b −1 γ (−∞, c i ]. From hypothesis 2 it follows that {x j } must have a sub-sequence {x j k } convergent to a point x ∈ C γ ∩ b −1 γ (−∞, c i ] and there exists an coray σ from x to γ that intersects b −1 γ (c i ) in q, hence S i is closed (see Figure 1). Next, we prove by contradiction that S i is open. Indeed, we assume contrary, that is, for q ∈ S i , suppose there exists a points sequence {y j } ⊂ b −1 γ (c i ) \ S i such that q := lim j→∞ y j . We denote by σ j and σ the corays sequence passing through y j and q, respectively. We denote by x the initial point on σ, and by our assumption x ∈ C γ ∩ b −1 γ (−∞, c i ]. Consider now a scalar δ > d(q, x) and the forward closed ball B + δ (q) := {p ∈ M | d(q, p) ≤ δ}. Obviously B + δ (q) is compact due to the forward complete hypothesis and Hopf-Rinow Theorem, and x ∈ B + δ (q). Let σ j denote a coray to γ emanating from y j = σ j (0). Since B + δ (q) is compact and y j / ∈ S i , we can extend backward σ j to some interval [t j , 0] with d(σ j (t j ), q) = δ. Any limit geodesic of the sequence {σ} j is a coray passing through q which contains x as an interior point, that is a contradiction (see Figure 2). It follows S i must be open set.
In other words, what we have proved up to this point is that for any point q ∈ b −1 γ (c i ), there exists a maximal coray, i.e. a coray emanating from a point x ∈ C γ , passing through q.
Using this we proceed to proving that b γ (−∞, c i ] is compact. We assume the converse, that is there exists a divergent sequence {x j } in b γ (−∞, c i ]. From our claim it follows that for each j there exists a coray σ j from x j that intersects b γ (c i ) in y j , and we extend σ j up to the point z j = σ j (0) ∈ C γ .
From hypothesis 2 of the Theorem, there exists a subsequence z j k of z j convergent to z and hence there exists a point y ∈ b −1 γ (c i ) such that lim j→∞ y j = y. Since x j is interior point of the b γ (c i )-segment σ j | [0,s j ] , it follows that there exists a point x interior to the b γ (c i )-segment σ| [0,s] , where y j = σ j (s j ) and y = σ(s). But this implies that the sequence {x j } cannot be divergent, that is a contradiction. Therefore, b γ (−∞, c i ] must be compact. ✷ The following lemma shows that our Theorem 3.1 is a special case of Innami's result in [In1]. Lemma 3.2 Let (M, F ) be a forward complete Finsler manifold and γ a ray in M. If C γ = ∅ is compact, then for all sufficiently large a ∈ R, the level set b −1 γ (a) is arcwise connected.
Proof. Since C γ = ∅ is compact we can choose a number a > max b γ (C γ ). Thus there does not exist a co-point of γ in b −1 γ [a, ∞). Choose any two points x and y in b −1 γ (a). By Lemma 2.3, there exists a continuous curve c in b −1 γ [a, ∞) joining x to y. Since C γ ∩b −1 γ [a, ∞) = ∅, we can get a curve in b −1 γ (a) joining x to y by deforming the curve c along the corays intersecting c. Therefore, the level set is arcwise connected. ✷ 3. For any a > b γ (p) the level sets b −1 γ (a) coincide with the forward spheres S + (p, a − b γ (p)).

Moreover, we have
Proof. 1. Since p ∈ C γ is isolated it means we can choose a small enough ε > 0 such that all corays with initial points in M \ S + (p, δ) do not intersect S + (p, ε), for some ε < δ, and all maximal asymptotes straight lines (if any) do not intersect S + (p, ε), where S + (p, ε) := {x ∈ M | d(p, x) = ε} is the forward sphere in (M, F ).
This implies that every maximal coray containing a point in S + (p, ε) must have p as initial point. Indeed, let us consider a convergent points sequence {p j } ⊂ S + (p, ε) such that lim j→∞ = p, and denote by σ and σ j the corays to γ from the points p and p j , respectively. By construction, the corays σ j will extend backward to a maximal coray with initial point z j , or to an asymptotic straight line to γ (we do not need to assume here backward completeness, one of these two cases will happen anyway). In the case σ j extends to a maximal coray, since p j is convergent to p, the coray σ j is convergent to p. But this means that σ extends backwards outside S + (p, ε) to an initial point z := lim j→∞ z j , contradiction with the fact that p ∈ C γ . The case of asymptotic straight lines is not possible either. It follows that S + (p, ε) is simply covered by maximal corays with initial point p and therefore these simply covers M.
From statement 1 proved above it follows that there exists a maximal coray through q with initial point p, that is contradiction with the hypothesis q ∈ C γ . Obviously, maximal asymptotic straight lines are not allowed either.
Indeed, let q ∈ b −1 γ (a), that is b γ (q) = a, and let us again consider the maximal coray from p through q. Then from Theorem 1.1 we have b γ (q) = b γ (p) + d(p, q), hence d(p, q) = b γ (q) − b γ (p) = a − b γ (p) = r, and therefore q ∈ S + (p, δ). ✷ From Theorem 1.2 statement 2 we obtain Corollary 3.4 Let (M, F ) be a forward complete non-compact Finsler manifold. If from each point of M there exists a unique coray to a ray γ, then C γ = ∅.
Remark 3.5 It would be interesting to obtain some geometrical conditions on the Finsler manifold (M, F ) such that all Busemann functions are everywhere differentiable. Since this topic requires more elaboration, we leave it for a future research.
We recall that an end ε of a non-compact manifold X is an assignment to each compact set K ⊂ X a component ε(K) of X \ K such that ε(K 1 ) ⊃ ε(K 2 ) if K 1 ⊂ K 2 . Every non-compact manifold has at least one end. For instance, R n has one end if n > 1 and two ends if n = 1. By definition one can see that a product R × N has one end if N is non-compact and two ends otherwise.
Here we prove Corollary 3.6 Let (M, F ) be a forward complete non-compact Finsler manifold.

2.
If M has at least three ends, then there are no differentiable Busemann functions on M.
Proof. 1. Since C γ = ∅, it follows that b γ is smooth everywhere and hence from each point there is a unique coray to γ. Thus, we can define the function ϕ : M → R × b −1 γ (0), p → ϕ(b γ (p), h 1 (p)), where h 1 (p) is the intersection point of the coray from p with the level set b −1 γ (0). Obviously this is a homeomorphism. 2. Due to statement 1 it follows that if b γ is differentiable, then M have at most two ends. Statement 2 follows by logical negation. ✷

Examples
Example 4.1 We start by recalling here a Riemannian example from [N1]. Consider R 3 to be the 3-dimensional Euclidean space with the canonical metric. In the xy-plane we consider the disk D of centre (1, 1, 0) and radius 1/2. We can erect now a half cylinder (x − 1) 2 + (y − 1) 2 = 1 16 , z ≥ 0.
(4.1) Moreover, we can now connect the points (x 1 , y 1 , 0) on the circle (x − 1) 2 + (y − 1) 2 = 1 4 to the points (x 2 , y 2 , z 2 ) on the circle (x − 1) 2 + (y − 1) 2 = 1 16 , z = 1 4 by a 4-th order algebraic curve such that we obtain a smooth surface made of the xy-plane and the cylinder attached to (1, 1, 0) and smoothed out by the algebraic curve. Obviously this is a complete Riemannian surface. Let the positive y-axis to be the ray γ and let us consider the curve c on S obtained by the intersection of the surface S with the plane x = 1, y ∈ (−∞, 1]. We consider a point P ∈ c, a divergent sequence of numbers t i , and consider a sequence of geodesic segments from P to γ(t i ), for each i. For i large enough there are two geodesic segments from P to γ i denoted by σ 1 i and σ 2 i and by taking the limit i → ∞ we obtain two maximal corays σ 1 and σ 2 from P to γ given by lim i→∞ σ 1 i and lim i→∞ σ 2 i , respectively. That is P ∈ C γ . One can see that any point on the curve c has this property and therefore one concludes that C γ = c.
Next, we will deform this Riemannian structure to a Randers metric on the same manifold. Denoting by h the restriction to the half cylinder (4.1) of the Riemannian metric constructed above, then we see that it is quite easy to construct a Finsler metric F = α + β of Randers type on S.
Indeed, we can regard the half cylinder (4.1) as a surface of revolution and consider the rotation around the straight line {x = 1, y = 1}. Obviously this is a parallel straight line with the z-axis, piercing the xy-plane in the point (1, 1, 0). More precisely, we consider the vector field W := (− 1 4 y, 1 4 x, 0) whose flow is given by where p = (x, y, z) is a point of the half cylinder (4.1). This is the rotation that leaves unchanged the parallels S ∩ {z = c} and twist the meridians x − 1 = 1 4 cos θ, y − 1 = 1 4 sin θ, z ≥ 1 4 along the half cylinder, here (x, y) belongs to the circle of centre (1, 1) and radius 1 4 . Obviously the twisted meridians are geodesics of the Randers structure F = α + β (on the half cylinder) obtained as solution of the Zermelo's navigation problem for navigation data (h, W ).
We will extend this Randers metric to a smooth Finsler metric defined on all surface S. Indeed, we consider , for z ∈ (0, 1 4 ) (0, 0, 0), for z = 0, on the smooth surface S, where p = (x, y, z) ∈ S. This is a smooth vector field on S.
The Randers metric F = α + β obtained as the solution of Zermelo's navigation problem for navigation data ( h, W ) , where h is the Riemannian metric on S, is a Finsler metric on S, that is Riemannian on S ∩ {z = 0}. It is easy to see that C γ = c with respect to this Finsler metric.
It is known that the cut locus of a point in a Riemannian or Finsler manifold M is a closed subset of M (see [BCS]). Moreover, we have shown in [TS] that the cut locus of a closed subset in M is not closed in M anymore. A natural question is if the set of co-points C γ is closed or not. The answer is given hereafter.
Example 4.2 We start again with a Riemannian construction obtained by the iteration of the Riemannian construction in Example 4.1 (see [N1]). Indeed, consider again R 3 and take a sequence of points D n in the xy plane given by D n = 2n + 1 2n(n + 1) , n, 0 and denote by γ the y-axis.
Next consider the sequence of disks in the xy plane with center D n and radius 1 2n(n+1) . As in the previous example we cut out this disk and smoothly connect the boundary of the disk with the half cylinder x − 2n + 1 2n(n + 1) 2 + (y − n) 2 = 1 16n 2 (n + 1) 2 , z ≥ 0.
We obtain in this way a smooth surface S with countably many ends. Let us denote by γ the positive y-axis. Then by the same arguments as in the the previous example, the co-points set C γ contains the sequence of curves c n obtained by intersecting the surface S with the planes x = 2n+1 2n(n+1) , y ∈ (−∞, n]. We consider now the set of points q n = ( 2n+1 2n(n+1) , λ, 0) for a fixed constant λ ∈ (0, 3 4 ), for instance λ = 3 8 will do. Obviously {q n } is a sequence of co-points of γ, i.e. q n ∈ C γ . At limit, one can see that lim n→∞ q n = q = (0, λ, 0) ∈ γ. But the ray from q to γ is subray of γ so it cannot be maximal. In other words, q / ∈ C γ . This means that C γ is not closed. We can construct a Raders metric from this Riemannian construction as we did in Example 4.1. Using again the navigation data ( h, W ) and same construction from Example 4.1 we obtain a Finsler metric of Raders type whose set of co-points C γ is not closed.