The conjugate locus on convex surfaces
The conjugate locus of a point on a surface is the envelope of geodesics emanating radially from that point. In this paper we show that the conjugate locus of generic points on convex surfaces satisfy a simple relationship between the rotation index and the number of cusps. As a consequence we prove the ‘vierspitzensatz’: the conjugate locus of a generic point on a convex surface must have at least four cusps. Along the way we prove certain results about evolutes in the plane and we extend the discussion to the existence of ‘smooth loops’ and geodesic curvature.geodesic and conjugate locus and Jacobi field and rotation index and evolute and convex
The conjugate locus on convex surfaces
The conjugate locus is a classical topic in Differential Geometry (see Jacobi , Poincaré , Blaschke  and Myers ) which is undergoing renewed interest due to the recent proof of the famous Last Geometric Statement of Jacobi by Itoh and Kiyohara  (Figure \@setrefcps, see also ,, ) and the recent development of interesting applications (for example , , , ). Recent work by the author  showed how, as the base point is moved in the surface, the conjugate locus may spontaneously create or annihilate pairs of cusps; in particular when two new cusps are created there is also a new “loop” of a particular type, and this suggests a relationship between the number of cusps and the number of loops. The current work will develop this theme, the main result being:
Let be a smooth strictly convex surface and let be a generic point in . Then the conjugate locus of satisfies
where is the rotation index of the conjugate locus in and is the number of cusps.
We will say more precisely what we mean by ‘generic’ and ‘rotation index’ in the next sections. It’s worth pointing out that the main interest in the rotation index is that, for regular curves, is invariant under regular homotopy ; however envolopes in general are not regular curves. Nonetheless the rotation index gives us a sense of the ‘topology’ of the envelope, see for example Arnol’d  (if the reader wishes we could perform a ‘smoothing’ in the neighbourhood of the cusps to make the envelope regular, see ). This formula puts restrictions on the conjugate locus, and in particular we use it to then prove the so-called ‘vierspitzensatz’:
Let be a smooth strictly convex surface and let be a generic point in . The conjugate locus of must have at least 4 cusps.
A central theme of this paper is the analogy between the conjugate locus on convex surfaces and the evolutes of simple convex plane curves; in Section 2 we focus on the latter and prove some results relating to rotation index, the distribution of vertices, and smooth loops. In Section 3 we focus on the conjugate locus and after some definitions we prove Theorems \@setrefthmrot and \@setrefthmvier and derive a formula for the geodesic curvature of the conjugate locus. Theorems 1-4 are new to the best of our knowledge, as is the expression for the geodesic curvature in Section 3. The proof of Theorem \@setrefthmvier is new and makes no reference to the cut locus, and the ‘count’ defined in Section 3 is a development of some ideas in . Throughout a will mean derivative w.r.t. argument.
Let be a ‘plane oval’, by which we mean a simple smooth strictly convex plane curve (we take the signed curvature of to be negative for the sake of orientations, see below and Figure \@setreffigevo), and let be the evolute to (the envelope of lines normal to , also known as the caustic). The evolute has the following well known properties (see Bruce and Giblin  or Tabachnikov ): will be a closed bounded curve (since ); consists of an even number of smooth ‘arcs’ separated by cusps which correspond to the vertices of (ordinary cusps if the vertices of are simple); and is locally convex, by which we mean for every regular there is a neighbourhood of which lies entirely to one side of the tangent line to at .
We use the local convexity to define a ‘standard’ orientation on as follows: at any regular point we move along such that the tangent line is to the left of (changing the definition but keeping the sense of Whitney , see the black arrows in Figure \@setreffigevo). Note this implies is oriented in the opposite sense to . After performing one circuit of the tangent vector will have turned anticlockwise through an integer multiple of ; we call this the rotation index of (at cusps the tangent vector rotates by , see DoCarmo ). Note: we use the term ‘rotation index’ in keeping with , , , however some authors use ‘index’ , ‘rotation number’ , , ‘winding number’ ,  or ‘turning number’ .
A segment of between two vertices and the corresponding arc of have total curvatures equal in magnitude but opposite in sign.
By ‘total curvature’ we mean the arc-length integral of the signed curvature. Let and be the arc-length parameters of and respectively. It is well known  that if is the radius of curvature of then the curvature of is and where is the unit normal to . Therefore
Let be a plane oval and its evolute. The rotation index of , denoted , is given by
where if the number of cusps of (or vertices of ).
The rotation index of is defined by where is the angle the unit tangent vector to makes w.r.t. some fixed direction, thus
where sums over the smooth arcs of and the cusps of are at . From Lemma 1 the first term on the right is just the total curvature of which is because is simple, smooth and oriented oppositely to . Therefore
This simple formula puts restrictions on the form the evolute of a plane oval can take, for example if the evolute is simple it can only have 4 cusps (see Figures \@setrefappics and \@setrefappics2 for examples). For the benefit of the next Section we will rephrase this result in more general terms:
It is useful to imagine the tangent line to rolling along the evolute in a smooth monotone way; this is due to the combination of local convexity of the smooth arcs and the ordinariness of the cusps. To make this more precise, let us define an ‘alternating’ orientation on by transporting the inward pointing normal to along the normal line to where it is tangent to (red arrows in Figure \@setreffigevo), and let be the (unit) tangent vector to w.r.t. this orientation. The oriented tangent line defined by makes an angle w.r.t. some fixed direction, but importantly this angle varies in a smooth and monotone way as we move along w.r.t. the standard orientation; in fact it changes by after one circuit of (this is because is simply the angle made by the normal to , see Figure \@setreffigevo; this slightly overwrought description is necessary for later sections).
The rotation index of on the other hand is defined by the tangent vector w.r.t. the standard orientation. Let be this (unit) tangent vector and the angle it makes w.r.t. the same fixed direction. Since at cusps then
where is the variation over the th smooth arc. On the arcs where then , and on the arcs where then so again . Thus
In fact this is the case more generally, in the absence of any generating curve or local convexity (without some care is needed for point 2):
Let be a closed bounded piecewise-smooth planar curve with the following properties:
has an even number of smooth arcs separated by ordinary cusps,
the alternating oriented tangent line completes a single clockwise rotation in traversing w.r.t the standard orientation.
Then the rotation index of satisfies .
It just happens that the evolute of a plane oval satisfies these conditions. Actually we can generalize to evolutes of plane curves with that are not simple: the term is just the rotation of the normal line to , so if is the rotation index of then
As an example consider the well known evolute of a non-simple limaçon, where (see Figure \@setrefappics in the Appendix).
While the evolute of a plane oval can have rotation index as high as we like and can be quite convoluted, we will show that any loops must have cusps on them; there can be no smooth loops. For this we use the support function , a useful notion surprisingly absent from many texts on Differential Geometry. Let be a point inside and fix a direction. The line tangent to at any point has an orthogonal segment to of length making an angle with the fixed direction. The -periodic function is called the support function (well-defined since is convex), and is recovered from as follows: for each ray from making an angle w.r.t. some direction draw the line orthogonal to the ray a distance from ; is the envelope of these lines. The radius of curvature of is simply , and hence a vertex of occurs at a point where . Since is a -periodic function we can represent it by a Fourier series, and the function has a Fourier series beginning at the second harmonic. The following is sometimes used in the proof of the 4-vertex theorem , we adapt it slightly.
Suppose is a -periodic function whose Fourier series begins at the second harmonic. Then any interval of length must contain a zero of .
Consider the functions and defined by the following differential equations:
By the Sturm comparison theorem  any interval between successive zeroes of must contain a zero of . As the zeroes of are a distance apart, the zeroes of are at most a distance apart. Requiring to be -periodic gives a Fourier series beginning at the second harmonic. ∎
Let be a plane oval, and a point on . There is a unique point on such that the tangent lines at and are parallel. The points and divide into two segments; there must be a vertex in each segment. In particular, if is the evolute of then a smooth arc of cannot contain two parallel tangents.
Let be on the normal to at (see Figure \@setreffigevo). The angle made by the orthogonal ray from to the tangent lines at and differs by . If there is no vertex between and then the derivative of the radius of curvature of would have no zero in this interval, but since is a -periodic function whose Fourier series begins at the second harmonic then by Lemma 2 it must have a zero in any interval of length , and the first statement follows. Now suppose and are two points on with parallel tangents (see Figure \@setreffigevo). If there are no cusps on the arc of between and then there is no vertex on the arc of between and , but since the tangents at and are parallel then the tangents at and are parallel and this contradicts the first statement. ∎
This ‘distribution’ of vertices/cusps can be formulated in a number of different ways:
Let be a plane oval and its evolute.
If and are two points on a smooth arc of then .
Let be a point on and the line normal to at . It is not possible for all the vertices of to lie on one side of .
It is not possible for to have a smooth loop, that is to say if and , then has a cusp in .
This is not to say we cannot construct a curve whose evolute has a smooth loop, or indeed a smooth ‘semicircle’ as point 1 above suggests, it is just that if we try to construct and close such a curve it will not be simple and convex (see Figure \@setrefappics). In the next Section we will show how many, but not all, of these results carry over to the conjugate locus on convex surfaces.
Let be a smooth strictly convex surface, and let be a point in . The conjugate locus of in , denoted , is the envelope of geodesics emanating from ; put another way, is the set of all points conjugate to along geodesics emanating radially from . We choose to be strictly convex so every geodesic emanating from reaches a point conjugate to in finite distance; indeed if is the Gauss curvature of then (again using Sturm’s comparison theorem, see ,) a conjugate point must be reached no sooner than and no later than (max/min w.r.t. as a whole). Following Myers , we make the following definitions: fixing a direction in , the (unit) tangent vector of a radial geodesic at makes an angle w.r.t. this direction and this radial geodesic reaches a conjugate point after a distance . We call the ‘distance function’ (another term being the ‘focal radius’, see ), and the polar curve it defines in the ‘distance curve’ (see Figure \@setreffigexp). The distance curve lies in the annulus of radii and is a smooth closed curve (star-shaped w.r.t. ); it is the exponential map of this curve (i.e. the conjugate locus) that is cusped (see Figure \@setreffigexp).
The following facts about the conjugate locus of a point are well known (see  or  for example): letting be the unit speed geodesic with and making an angle w.r.t. some fixed direction in , we define the exponential map as . The Jacobi field satisfies the Jacobi equation, which we can write in scalar form as follows (in general everything depends on but for brevity we only denote the functional dependence where necessary): letting be the unit tangent and normal to we write and we can take w.l.o.g. (see ); then satisfies the scalar Jacobi equation
where is evaluated along the radial geodesic and the partial derivatives are to emphasise that Jacobi fields vary along radial geodesics but also from one radial geodesic to another. With initial conditions the first non-trivial zero of is a point conjugate to . The distance curve is therefore defined implicilty by , and the conjugate locus in is . Letting denote this parameterisation of , then
When is stationary then is not regular, it has a cusp (see Figure \@setreffigexp).
It is worth noting that the last expression tells us the following: is the arc-length parameter of ; the length of an arc of is simply the difference between consecutive stationary values of and since lives in an annulus this can bound the total length of ; if we give an alternating signed length to then its total length is zero; and finally the ‘string’ construction of involutes carries over to conjugate loci (at move along a tangential geodesic a certain distance , at move along a tangential geodesic a distance ; for some the involute will be a point, ).
In , the author derived an expression for the local structure of the conjugate locus by considering the higher derivatives of the exponential map, and we reproduce the formula here: let be a point on the conjugate locus corresponding to and suppose has an singularity at , by which we mean the first derivatives of vanish but not the th. Then the leading behaviour in the parameterization of the conjugate locus at is
We will return to the term when we look at the geodesic curvature of (see (\theequation)), for now we just need that . From this expression we see that if (i.e. an singularity of ) then locally is a parabola opening in the direction of if and if (i.e. at regular points is locally convex), and if (but , an singularity) then has an ordinary (semicubical parabola) cusp. We will describe a point in as ‘generic’ if the distance function has at most singularities.
It is well known that the rotation index of a curve on a surface can be defined by that of its preimage w.r.t. the coordinate patch of the surface which contains the curve, indeed Hopf’s umlaufsatz does precisely that (see  or ). It is also well known that, with regards regular homotopy, every closed curve on a sphere has rotation index 0 or 1 . For our purposes this is losing too much information about the curve, so (following Arnol’d ) we define the rotation index of the conjugate locus as that on the surface with the point removed, which is equivalent to projecting the curve into a plane and finding the rotation index of that curve; we will essentially show this projected curve satisfies Theorem \@setrefbnog.
At the risk of confusing the reader we orient in the following way: at we move in such a way that the radial geodesic from which has conjugate point at lies to the right of w.r.t. the outward pointing normal of (this uses the local convexity of ; alternatively increases in a clockwise sense when looking down on , see Figure \@setrefcps); this is so the orientation of the projected curve agrees with the previous Section, and so on the ellipsoid has . Also, if a radial geodesic from has conjugate point at , we will refer to the portion as a ‘geodesic segment’.
Let be a smooth strictly convex surface and let be a generic point in . Then the conjugate locus of satisfies , where is the rotation index of the conjugate locus in and is the number of cusps.
We already know the local structure of the conjugate locus of a generic point in : smooth arcs separated by ordinary cusps. Let be the projection defined as follows: radially project from to the unique supporting plane of parallel to the one at . Since is smooth and convex this projection is a homeomorphism. Letting , then is a piecewise-smooth curve with an even number of smooth arcs separated by ordinary cusps. If, on completing one circuit of , the tangent line turns through , then from Section 2 we have where is the rotation index of in (and hence the rotation index of in ) and is the number of cusps to (and also ). The only thing left to show is that .
Let be a small geodesic circle in centred on with radius . will be a simple curve in which approaches a (large) circle as . The projection under of a geodesic segment gives a semi-infinite curve in which meets at a unique point. For each , we identify the tangent line to at with this point of intersection of and the projected geodesic segment . This identification is one-to-one and continuous, and since we traverse once in rotating about then the tangent line to makes one rotation and hence . A simple diagram will show with the orientations defined previously.
An immediate corollary to this theorem would be an alternative route to proving the Last Geometric Statement of Jacobi: if we could show the conjugate locus of a generic point on the ellipsoid has rotation index 1 then there must be precisely 4 cusps.
In this section we will describe a way of assigning a positive intger to the regions of (developing ideas in ) in order to give a convenient method of calculating the rotation index of . We will use this, and the results of the previous section, to prove the ‘vierspitzensatz’: that the conjugate locus of a point on a convex surface must have at least 4 cusps. In a footnote, Myers  refers to Blaschke  (in German) who in turn credits Carathéodory with a proof of this statement (see also p.206 of ).
We define on the complement of in the following (which we shall refer to as the ‘count’):
as the number of geodesic segments through the point (we also exclude as the count is infinte there). The count is constant on each component of , it is never equal to zero (see ), it is equal to 1 in the component of containing (see Figures \@setreffigeul and \@setrefcps for examples) and it is preserved under the projection of the previous Subsection. As passes from one region of (or ) to another the count changes according to the rules shown in Figure \@setreffigcount; we know this from the locally convex structure of described in (\theequation).
We note another way of defining the count: the exponential map of the interior of the distance curve will cover and some points will be covered more than once; the number of times a point is covered by this map is . This alternative formulation may be useful in Gauss-Bonnet type constructions .
We can use the count to describe a method of calculating the rotation index of due to McIntyre and Cairns . In their paper they prove the following (we have adapted their statement to fit the current situation):
Theorem (McIntyre and Cairns).
Assigning a number to each region of according to the count, let be the union of the closure of the regions numbered , and the Euler characteristic of . Then the rotation index of in is
The example in Figure \@setreffigeul is typical: the regions are usually discs or the disjoint union of discs, and hence have Euler characteristic . If this were the case, the rotation index would be and hence could not be 2 via Theorem \@setrefthmrot. Nonetheless, it is also possible that the are discs less discs, i.e. they could have ‘holes’:
For every hole in a region , there is a disc.
Suppose there is a region which has a hole; the count inside the hole must be (see the right of Figure \@setreffigcount for an example). The boundary of the hole must contain points of self-intersection of , as is connected, and the interior is star-shaped with repsect to any point in the interior, due to local convexity of . If there is a point of self-intersection then there must be a sub-region of with count . Either this region is a disc, in which case we are done, or it also has a hole. If it has a hole, then applying the same reasoning there must also be a region with count , and so on. Eventually there must be a region without a hole, which is the disc we require. ∎
Let be a smooth strictly convex surface and let be a generic point in . The conjugate locus of must have at least 4 cusps.
According to Theorem \@setrefthmrot, we must rule out the case of having rotation index 0 (if is generic then must have an even, and non-zero, number of stationary points which rules out ). But by Lemma 3 the sum of the Euler characteristics of the regions must be , and hence the rotation index cannot be 0 and the number of cusps cannot be 2. ∎
The rules of the count put other restrictions on , for example: the conjugate locus cannot have any ‘necks’, by which we mean a point of intersection separating the conjugate locus into two non-overlapping components, and for any regular the locally convex side of (the side containing the geodesic segment ) cannot be the ‘outside’ (the component containing ). Continuing in this vein and in analogy with evolutes of plane ovals it might seem reasonable to expect a ‘no smooth loops’ statement to follow; however a counterexample would be the right of Figure \@setrefappics2 which satisfies all the requirements of the count and Theorem \@setrefthmrot and yet has a smooth loop.
Looking more deeply into the problem, suppose we view , the distance curve in , as the radius of curvature function of some plane curve . From Theorem \@setrefthmvier has at least 4 stationary points and by the converse of the 4-vertex theorem  is therefore a simple closed curve; moreover it is strictly convex since . The evolute of , let’s call it , is a sort of dual to the conjugate locus, . Both and have the same number of cusps, and thus the same rotation index (in and respectively), and has no smooth loops (by Corollary 1). This however is not enough to ensure cannot have any smooth loops, we give a counter example:
Up to now we have been studying the ‘first’ conjugate locus, where is the first non-trivial zero of . We can extend this to consider the th conjugate locus of a point , denoted , by simply letting be the th non-trivial zero of . Now each distance curve will have the same properties as described in the previous paragraph, however (as suggested in  and shown in Figure \@setrefcps) the conjugate locus can have smooth loops. Indeed the annulus in that the distance curve lives in gets wider as increases and thus the lengths of the arcs of can be very long. Nonetheless, it seems likely to us that there can be no smooth loops in the first conjugate locus of a generic point on a smooth strictly convex surface, or to put it another way, an arc of cannot intersect itself.
As it is relevant for this discussion, but seems to be missing from the literature, we derive an expression for the geodesic curvature of the conjugate locus. Recall from (\theequation) that the unit tangent vector to is , so the derivative is
since by definiton of a geodesic. If then . Now letting be the arc-length of the Frenet equation for geodesic curvature  gives
This explains the appearance of the term in (\theequation). We note that is never zero (and hence is never zero): since is defined by then and the Jacobi equation (\theequation) would imply . Rather since and is the first non-trivial zero of we have . The formula applies equally well to , if we simply adapt to be the th non-trivial zero of (note the sign of will alternate from to ). Another expression for the geodesic curvature is found by differentiating w.r.t. (see the ‘focal differential equation’ in ) to give (see  for a definition of ).
Note the sign of in (\theequation) alternates between positive and negative depending on whether is decreasing or increasing on an arc of , whereas using the orientation defined before Theorem \@setrefthmrot in Section 3.1 is strictly positive; this is because (\theequation) is derived w.r.t. the tangents to radial geodesics which define an alternating orientation on , similar to that described in Section 2.1.
While the geodesic curvature in (\theequation) diverges when is stationary (as we already knew), the total curvature
does not depend explicitly on . On the sphere , so on a perturbed sphere we might expect the total curvature of an arc of the conjugate locus to be close to the difference in the values of at which is stationary which suggests an upper bound on (in line with Corrolary 1), however moving away from the sphere it seems the arcs of the conjugate locus can become very curved indeed.
We cannot simply draw a plane curve with an even number of arcs and cusps and claim it to be the evolute of a plane oval; there are restrictions both geometrical (local convexity, sum of the alternating lengths vanishing) and topological (rotation index, no smooth loops), as we have shown. In the same way we have shown that there are restrictions on the geometry and topology of the conjugate locus of a generic point on a smooth strictly convex surface, and these restrictions lead to structure which can be exploited.
There are a number of generalizations of the results in this paper which are worth considering. We have already alluded to the ‘higher’ conjugate locus , and it is possible there is a formula generalizing (4). Also we remind the reader that the conjugate locus of is the envelope of geodesics normal to the geodesic circle centred on (in this context the term ‘caustic’ may be more appropriate). For small radius the geodesic circles are simple, and it is possible the caustics of simple curves on convex surfaces also satisfy (4) whereas nonsimple curves lead to (\theequation) (possibly with the requirement the curves themselves have ), however initial experiments suggest this context to be surprisingly complicated. Finally we have only considered convex surfaces however it is likely expressions such as (4) carry over to spheres with as long as the regions of negative Gauss curvature do not contain any closed geodesics (for example the spherical harmonic surfaces of , as opposed to the ‘dumbell’ of ).
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