Isoperimetric inequalities for wave fronts and a generalization of Menzin’s conjecture for bicycle monodromy on surfaces of constant curvature
The classical isoperimetric inequality relates the lengths of curves to the areas that they bound. More specifically, we have that for a smooth, simple closed curve of length bounding area on a surface of constant curvature ,
with equality holding only if the curve is a geodesic circle. We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss “bicycle curves” using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unit segment that can move so that it remains tangent to the trajectory of point (the rear wheel is fixed on the bicycle frame), as discussed in , , and . We extend this definition to a general Riemannian manifold, and concern ourselves in particular with bicycle curves in the hyperbolic plane and on the sphere . We prove results along the lines of those in  and resolve both spherical and hyperbolic versions of Menzin’s conjecture, which relates the area bounded by a curve to its associated monodromy map.
We will use the standard model of the sphere, , as an embedded surface in , consisting of all position vectors with euclidean magnitude 1. There are several models of the hyperbolic plane, , but we will find it convenient to use the hyperboloid model 111For more information about hyperbolic geometry, see  or .. In the hyperboloid model, the hyperbolic plane is realized as the set of points in whose position vectors have coordinate and norm -1 with respect to the quadratic form
This corresponds to the positive sheet of the hyperboloid of two sheets embedded in . Restricted to the tangent space of the hyperboloid, the above quadratic form is non-negative, and therefore defines a genuine Riemannian metric on .
The primary object we will be working with in this paper is called a wave front.
Let be a curve. We say that is a wave front if it is piecewise smooth and its only singularities are (semicubical) cusps.
1.2. The Isoperimetric Inequality
As our paper deals with a generalization of the classical isoperimetric inequality to wave fronts, let us first discuss the history of the problem. The classical isoperimetric inequality relates the length of a plane curve to the area that it bounds. More precisely we have that for a smooth, simple closed curve of length bounding area on a surface of constant curvature ,
with equality holding only if the curve is a geodesic circle.
T.F. Banchoff and W.F. Pohl  generalized the isoperimetric inequality to non-simple curves in the euclidean plane. In the case of a non-simple curve one must redefine the notion of area. is replaced by the sum of the areas into which the curve divides the plane, weighted by the square of the winding number:
where is a smooth immersion and is a winding number of with respect to the point .
J.L. Weiner generalized this result to smooth immersions of a circle into the 2-sphere, in  (which we use to prove some later results). Using the Gauss-Bonnet Theorem, the classical isoperimetric inequality can be written as
where , with being the geodesic curvature and the element of arc length. This inequality makes sense for immersions as well as embeddings. In , Weiner shows that (1) holds for smooth immersions regularly homotopic to a circle traversed once.
E. Teufel made a similar generalization for smooth immersions of a circle into the hyperbolic plane, in . Teufel showed that
Here is a smooth immersion of a circle into the hyperbolic plane , is the winding number of with respect to the point , and is the area element of at the point . In all three geometries the equality holds only if the curve traverses a geodesic circle a number of times in the same direction.
One of the goals of this paper will be to generalize the isoperimetric inequality to wave fronts. In , Martinez-Maure has obtained similar inequalities for euclidean wave fronts, but we concern ourselves mainly with the hyperbolic plane and the sphere.
2. On The Sphere
2.1. Spherical Areas
Because the euclidean and hyperbolic planes share the property that all closed forms are exact, we have an unambiguous way of defining area. In the case of the plane, we can say that the area bounded by an oriented, simple curve is the line integral
Note that this integral can be rewritten as some integral with respect to the arclength element of the curve, . By Stokes’ theorem, this is the same as integrating over the region which has as its boundary (with the induced orientation). We can extend this definition to wave fronts by signing the arclength element so that it changes when passing through cusps.
Due to the topology of the sphere, however, we do not have that its area form is the exterior derivative of any form. A way to get around this is to define a particular 2-chain which has a given curve as its boundary, and declare the area bounded by the curve to be the integral of the area form over this chain. This is the approach taken in , but we will concern ourselves with a more naive notion of area. Our definition of area will be restricted to a small class of curves, and is motivated by the Gauss-Bonnet Theorem.
For a convex, simple, smooth spherical curve , we define the area bounded by or as
where is the geodesic curvature of and the curve is oriented so that is positive. In this case, we say that is properly oriented.
We also need a notion of algebraic length, defined for oriented, co-oriented wave fronts on any surface. Note, if we have such a wave front, the orientation and co-orientation together induce a sign on the arclength element of the curve. This is done by signing the arclength element with the sign of the frame formed by the co-orientation and orientation.
Let be an oriented, co-oriented wave front, we define the algebraic length of , , to be
where is the element of arclength, signed as described above.
The approach taken in  by Arnold is significantly more complicated, but it allows us to make a reasonable definition of the area bounded by a much wider variety of curves, including those with cusps. The construction of this area (which we will denote by to avoid ambiguity) is analogous to using the winding number to define area in the plane, and is as follows (more specific details can be found in , we present only the basics).
For a closed, oriented, co-oriented spherical wave front , we have that divides the sphere into distinct regions. For each region, we pick a point (not on ) in this region and do stereographic projection using this point as the point at infinity. This turns into a planar curve, whose normal vector has a winding number (the number of turns it makes around the unit circle after normalization, which is dependent only on the choice of region). To each region, we assign the coefficient .
The characteristic 2-chain of is the formal sum of the regions that divides the sphere into with coefficients computed in the manner described above. Note that is the boundary of this 2-chain, by a result in .
Using this construction, we are now in a position to define a type of area on the sphere.
Let be the characteristic 2-chain of a wave front . Then the area of the characteristic 2-chain of or as the integral of the spherical area form over this 2-chain. That is, if is the spherical area form,
With this definition of area, we have a version of the Gauss-Bonnet Theorem (taken from ) which will be useful for us later.
For an oriented, co-oriented front with geodesic curvature , we have
where is the signed element of arclength, determined by the co-orientation of .
Given an oriented, co-oriented front on the sphere, we can obtain the dual of , denoted , by moving every point of a distance of along the great circle in the direction of the co-orientation.
We have that taking the dual of a curve turns double tangent points into self-intersections and turns cusps into inflections, as seen in Figure 1. As well, from  we have the following relations between the area and length of a curve and its dual.
Let be a closed, oriented, co-oriented wave front on the sphere. Then we have that
2.2. A Spherical Isoperimetric Inequality
Using the above, we are in a position to prove a spherical version of the classical isoperimetric inequality for wave fronts.
Let be a wave front without inflection points and having an even number of double-tangent points. Then
3. In the Hyperbolic Plane
3.1. A Hyperbolic Isoperimetric Inequality
Now we seek to prove a similar isoperimetric inequality for hyperbolic wave fronts, using an approach similar to those taken by Anisov in  and . Analogously to the approach we took on the sphere, we are going to introduce a new function where is the geodesic curvature of . We are going to let be defined for simple curves as the regular area of . From the Gauss-Bonnet theorem, we have that .
Let be a closed, oriented, co-oriented and horocyclically convex wave front (i.e. its geodesic curvature has magnitude that is everywhere ) with turning number 1 in . Let , where the arc length changes sign at every cusp, and let be its algebraic length. Then we have
with equality iff is a geodesic circle.
Consider the family of equidistant fronts of . Let and be and respectively. We are going to show that and (where denotes differentiation with respect to ). To do this we are going to consider the contribution from each smooth arc.
Let be a smooth arc of the curve , and let be its corresponding arc in . Let be the area of the region between the two arcs, as in Figure 2. From the Gauss-Bonnet theorem we have that
By looking at an infinitesimal change in , we can see that , and by differentiating both sides we obtain
To show that we are going to apply a local Steiner formula to a polygonal approximation of the curve by geodesic arcs and equidistant arcs as in Figure 3.
Since the circumference of a circle of radius in the hyperbolic plane is we see that
We also have that
Ignoring the terms in of higher order we see that
Taking the limit as the number of vertices of goes to infinity we get that . Therefore .
Now we want to find the explicit formula for and . To do this we are going to solve the system of differential equations
from which we get that
where and . We now note that the quantity is independent of the value of . The following lemma shows that for a very large value of , is simple and smooth, so that we can apply the classical isoperimetric inequality.
Let be a closed, horocyclically convex wave front with turning number one and let be the family of equidistant fronts of . Then, for sufficiently large , is smooth and simple.
We use the hyperbolic support function of Leichtweiß, found in . The hyperbolic support function is a periodic function, defined for each wave front. This function characterizes the curve, and is unique up to the choice of origin and the original direction. Note that this is a hyperbolic generalization of the support function that exists for euclidean curves (see , for example). We have that the hyperbolic support function is defined only for curves whose geodesic curvature is greater than or equal to one (are horocyclically convex), with turning number one. Adding a constant to the support function generates an equidistant curve to the original. The magnitude of the curvature is given by
where is the hyperbolic support function. The cusps correspond to the value of the denominator being zero. The claim is that if we add a sufficiently large constant to then the the geodesic curvature will be always greater than zero and less then . Our result then follows, because under those conditions we have a smooth curve with everywhere positive curvature and turning number one, which implies that the curve is simple.
First we show that for a large the denominator is not zero.
The quantities and do not change when a constant is added to the function. Clearly both and can not be zero. The only case where the denominator could be zero is if both and are non zero and have opposite signs. In this case if we add a large constant to , the term dominates and the denominator will not be zero. The same argument works for the numerator of the fraction. Since is periodic there exists a constant such that both the numerator and the denominator of the expression of the geodesic curvature never vanish.
Let be the point at which is a simple, smooth curve. By the classical isoperimetric inequality, we have that
Using the Gauss-Bonnet theorem and the fact that is simple we get that
But since the quantity on the left is independent of we see that
If is a geodesic circle, then the classical isoperimetric inequality applies, and the inequality in the statement of the theorem becomes an equality. Conversely, if we have equality for then some equidistant of is a geodesic circle, which implies that itself is a geodesic circle. ∎
4. An Application to Bicycle Curves
We now discuss the subject of bicycle curves, where isoperimetric inequalities for wave fronts will prove to be a useful tool.
We first define our model of bicycle motion. The bicycle frame is represented by a geodesic line segment of constant length , while the back and front wheels are represented by and respectively. The path of the front wheel is restricted so that is always tangent to the velocity of the back wheel, . This is illustrated in Figure 4.
An important construction that we will consider is the monodromy map associated to a particular front wheel curve.
Let be a curve representing the motion of the front wheel. Up to choice of relative initial position of the back wheel, the path of the back wheel is determined by . The monodromy map associated to or is the function that sends each choice of starting position for the back wheel to its terminal position.
Finally, let’s establish the notation that will be used for the remainder of the paper. The space that each result applies to ( or ) will be noted in parentheses. By denote the segment representing the bicycle frame from back wheel to front. will denote the path of the front wheel and the path of the back wheel. Let be the arc length parameter for and the arc length parameter for , and the respective curvatures. Let be the angle between and , which can be parameterized by either or . Finally, will denote the algebraic length of the curve .
4.1. General Results
In this section we develop some general results about bicycle curves in and that will allow us to describe how they evolve. These results will primarily concern themselves with the relationships between properties of the front and back wheel curves, which will be essential to establishing our later results about the monodromy map and our reformulation of Menzin’s conjecture.
Theorem 4.1 ().
Let be the function that sends a rear wheel curve to the corresponding front wheel curve for a bicycle of length . The condition is equivalent to the differential equation on the function :
We will denote by the tangent vector to at and by the tangent vector to at , as in Figure 5.
First we write down an expression for and use it to obtain an expression for :
Note that . We can use this to find the derivative :
We can also obtain an expression for :
Now, and are parallel, so their cross product must be 0:
Since these three vectors are orthogonal and non-zero, all of A, B, and C must be equal to 0. This gives us that either
This system cannot be satisfied for any pair of and , however, so the 1st claim of the theorem is proven.
To establish the 2nd claim, we write the expression for as above and take its magnitude. Using the differential equation that we have just derived, we substitute and simplify, which yields the result. ∎
Parameterizing by , we have that
We parameterize everything below by . Let denote the orientation of the rear wheel ( if the same direction as and otherwise). We have that
We can also write
Plugging in our formula for from above and equating the two expressions we find that
Substituting we see this is solved for
The above results on the sphere have parallels in the hyperbolic plane, which we prove below. Note that they are identical to the spherical results up to the replacement of trigonometric functions with hyperbolic trigonometric functions.
Theorem 4.3 ().
The condition is equivalent to the differential equation on the function :
First we write down an expression for and use it to obtain an expression for :
We have that is a unit tangent vector in the direction of the geodesic connecting and , so we can use the standard parameterization of geodesics in the hyperboloid model to obtain an expression for :
By the hyperbolic Frenet-Serret formulas, we have that
Using this, we can find an expression for
As in the spherical case, we can find by transporting along the geodesic :
We must have that and must be parallel (in the Euclidean sense), and so we can equate their Euclidean cross product to 0, giving us
We have that and are linearly independent, so must all be 0. This gives us that either
This system cannot be satisfied for any pair of and , however, so we are done. ∎
Corollary 4.4 ().
Parameterizing by , we have that
We parameterize everything below by . Let denote the orientation of the rear wheel ( if the same direction as and otherwise). Note
Now, by denote the Lorentz cross product given by
After substituting we get
Now, we can find by parameterizing the arc of the bicycle starting at , taking the derivative at , and flipping it to get:
However, it can be verified that and , so
We also have that where is the Minkowski norm. This gives us:
Substituting in for and equating our two expressions we find
Which gives us that . ∎
4.2. The Monodromy Map
Having established differential equations that describe the bicycle motion on the sphere and in the hyperbolic plane, we are now in a position to talk about the monodromy map associated to a front wheel curve .
Note that in the euclidean plane we have the ability to identify circles centered at different points of via parallel translation. On the sphere or in the hyperbolic plane, this is not possible. Instead, we identify circles at different points of so that the velocity vector makes an angle of 0 degrees. With this convention, we can define the monodromy map as above, and we see that the monodromy map is always a Möbius transformation, but let us first make sense of that statement.
A Möbius transformation is a fractional linear map , of the form
for some . In matrix form:
We have that the orientation-preserving isometries of in the upper half-plane model are those Möbius transformations with real coefficients and such that . These correspond to the group of real matrices with determinant 1. The isometries extend to the real line and the point at infinity, which can be identified with or . This gives us a way for Möbius transformations to act on , which is the domain and range of our monodromy map.
Let be a front wheel bicycle curve. Then the monodromy map associated to is a Möbius transformation.
First move from the coordinate to the projective coordinate . We can then rewrite the differential equation (5) as
We then have that, treating as a vector field and with being a unit vector on ,
so that the vector field is a combination of the vector fields , , and with -dependent coefficients. These vector fields generate the Lie algebra of , which acts on via Möbius transformations. So is a transformation whose infinitesimal action is the same of that of a Möbius transformation, implying that the monodromy is in fact a Möbius transformation. ∎
The Möbius transformations that we are dealing with are identified with isometries of the hyperbolic plane, which come in 3 types: elliptic, parabolic, and hyperbolic. These transformations have 0, 1, and 2 fixed points on the circle at infinity, respectively. Note that if the front wheel curve is closed, a fixed point of its associated monodromy map corresponds to a closed rear wheel curve.
Theorem 4.6 (Both).
For sufficiently small values of , the monodromy map is hyperbolic.
Theorem 4.7 (Both).
Let have a fixed point and let be the closed curve of the rear wheel corresponding to . Then we have
We consider the spherical case first. If has a fixed point, we have that there exists an that is an periodic solution to (5). Consider an infinitesimal perturbation of this solution, . First, we have that . For to satisfy (5), we must have
Using the fact that is a solution to (5) and that is small, we get
We then have that
Corollary 4.8 (Both).
is parabolic if and only if the algebraic length of is 0.
In the parabolic case, , which implies that Conversely, if , then , which implies that is parabolic, as the derivatives at fixed points are reciprocal to each other. ∎
Corollary 4.9 (Both).
In the parabolic case, has cusps.
If a closed curve is to have 0 length, there must be arcs with opposite parity. This implies that such a curve must have cusps, as this is the only way to get arcs with different parities. ∎
On the sphere, we have the notion of a derivative curve as discussed in . Given a spherical wave front we associate to it the curve obtained by moving every point a distance of in the direction tangent to the curve at that point. This is the same as using as a back wheel curve for a bicycle of length . By a result in , we have that the derivative curves are the same for any equidistant front of , and so the produced as the front wheel curve has a whole family of back wheel trajectories that generate it. This implies that the monodromy map associated to is the identity, as it has an infinite number of fixed points (corresponding to the family of equidistance fronts of that generate it).
4.3. A Spherical Menzin’s Conjecture
In , Levi and Tabachnikov prove an old conjecture of Menzin for bicycle curves in the plane. The conjecture states that if a closed, convex front wheel curve bounds area greater than (the area of a unit circle) then the associated monodromy map is hyperbolic (for , other choices of simply scale the value ). On the sphere, a circle of radius has area , so it seems plausible that the monodromy map would be hyperbolic in the case that a closed, convex front wheel curve bounds area greater than this. This conjecture is in fact true, as the following shows.
Theorem 4.10 ().
Let be a closed, convex curve, oriented properly. Then if is greater than , the corresponding monodromy map is hyperbolic.
We argue by contradiction, assuming that and is not hyperbolic. We need to show that
Let the length of the bicycle vary as (with from 0 to 1). For small , we have that is hyperbolic, by Theorem 4.6. This cannot continue for all , however, so there must be a such that is parabolic. Let be the closed back wheel curve corresponding to the fixed point of for . Because we have that , so to establish the above inequality it suffices to show that
We now prove a lemma relating the curvature of the back and front wheels that will allow us to simplify this inequality.
Letting be the curvature of the front wheel and the curvature of the back wheel, we have
where integrals are taken over their respective curves and the elements of arclength are signed. Equivalently,
We first rewrite the integral of the curvature of using (7)
Note that is signed to reflect changes when passing through cusps. Integrating both sides of (5), we get
As is periodic, the first term on the left hand side is 0, so combining this with the above we have
To complete the proof, we need the following lemma.
The rear wheel curve has no inflection points and an even number of double tangent points.
First we show that has an even number of double tangent points. Consider the family of dual curves , as defined in 6. When is small we know that is convex, and therefore is smooth and has no inflections. As varies changes by a regular homotopy, since has no inflection points for any values of . Then is regularly homotopic to a circle traversed once for all and thus has an even number of intersections. Therefore has an even number of double-tangent points.
Now we show that has no inflection points. First, we compute an expression for in terms of . Using (7), we get
Placing this expression into (5), we obtain
Suppose that there is an inflection point of . Consider the family of fronts , for varying , and let be the least value of for which . For slightly greater than , there is a dimple in that causes to change sign twice, starting from a positive value. Then there exists an such that and , and by the above equation we have that . However, because is convex, we must have that , and so this is impossible. ∎
4.4. A Hyperbolic Menzin’s Conjecture
We now prove a version of the previous theorem in the hyperbolic plane. Because of some peculiarities of the hyperbolic plane, the hyperbolic Menzin’s Conjecture is proven only for horocyclically convex curves (i.e. hyperbolic curves with curvature ).
Let be a closed, horocyclically convex curve in the hyperbolic plane. Then if is greater then , the corresponding monodromy map is hyperbolic.
We argue by contradiction, assuming that and is not hyperbolic. We need to show that
Let the length of the bicycle vary as . For small , we have that is hyperbolic. There exists a such that is parabolic. Let be the closed back wheel curve corresponding to the fixed point of for . Because we have that , so to establish the above inequality it suffices to show that
we need to show
By virtue of the generalized isoperimetric inequality proven above we have
We also have that , as is parabolic. Putting it all together, we get that
which gives us the desired contradiction. ∎
5. Remarks and Conclusion
Our initial goal in this paper was to generalize to surfaces of constant curvature the results in  on bicycle monodromy in the Euclidean plane. Largely we have succeeded, providing analogs in both and to all of their major results. In addition, our method of attack led to the development of a useful tool - the isoperimetric inequality for wavefronts. However, there is one aspect of Menzin’s conjecture in that we have not been able to satisfactorily address. In both and we require convexity for the front wheel curve, but in our version of the conjecture requires the front wheel curve to be horocyclically convex. The question is then whether this requirement is necessary or just an artifact of our proof technique. Our guess is the former, but we lack a proof of the fact or an example of a convex but not horocyclically convex front wheel path in the hyperbolic plane bounding the correct amount of area but without hyperbolic monodromy. This question will need to be resolved before we can really close the book on Menzin’s conjecture in
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