Homotopies of Curves on the 2-Sphere with Geodesic Curvature in a Prescribed Interval
For , let denote the set of all closed curves of class on the sphere whose geodesic curvatures lie in the interval , furnished with the topology (for some . In 1970, J. Little proved that the space of closed curves having positive geodesic curvature has three connected components. Let (). In this thesis, we show that has connected components , where
and contains circles traversed times (). The component also contains circles traversed times, and contains circles traversed times, for any . In addition, each of is homotopy equivalent to (). A direct characterization of the components in terms of the properties of a curve and a proof that is homeomorphic to whenever () are also presented.
figurechapter \counterwithoutequationchapter \authorRZühlke, Pedro \orientadorNicolau Corção Saldanha \orientadorRSaldanha, Nicolau C. \titlebrHomotopias de Curvas na Esfera com Curvatura Geodésica num Intervalo Dado 10 \mesSeptember \ano2012 \cityRio de Janeiro \CDD510 \departmentMatemática \departamentoMatemática \programMatemática \programbrMatemática \schoolCentro Técnico Científico \universityPontifícia Universidade Católica do Rio de Janeiro \uniPUC-Rio \jury \jurymemberCarlos Gustavo Tamm de Araújo MoreiraInstituto Nacional de Matemática Pura e Aplicada (IMPA) \jurymemberCarlos TomeiDepartamento de Matemática – PUC-Rio \jurymemberJairo da Silva BochiDepartamento de Matemática – PUC-Rio \jurymemberPaul Alexander SchweitzerDepartamento de Matemática – PUC-Rio \jurymemberRicardo Sá EarpDepartamento de Matemática – PUC-Rio \jurymemberUmberto Leone HryniewiczInstituto de Matemática – UFRJ \schoolheadJosé Eugênio Leal \curriculo \agradecimentosI thank prof. Nicolau C. Saldanha for his support, patience and immense generosity. All of the results in this work bear his influence in some form. I feel privileged to be his student and friend. I also thank profs. Alexei N. Krasilnikov and Paul A. Schweitzer, S.J., for the kindness and generosity with which they have always treated me. Without the help of all three, I would hardly have obtained a PhD degree. During the last few years I was partially supported by scholarships (from CNPq and CAPES); I would like to thank everyone who worked to make them available to me. \keywordsbr \keyCurva \keyCurvatura \keyGeometria \keyHomotopia \keyTopologia \tituloHomotopies of Curves on the 2-Sphere with Geodesic Curvature in a Prescribed Interval \abstractbr Para , seja o conjunto de todas as curvas fechadas de classe na esfera cujas curvaturas geodésicas estão restritas ao intervalo , munido da topologia (para algum ). Em 1970, J. Little provou que o espaço de curvas fechadas com curvatura geodésica positiva possui três componentes conexas. Sejam (). Nesta tese, mostramos que possui componentes conexas , onde
e contém círculos percorridos vezes (). A componente também contém círculos percorridos vezes, e contém círculos percorridos vezes, para qualquer . Além disto, são todos homotopicamente equivalentes a (). Também são exibidas uma caracterização das componentes em termos das propriedades de uma curva e uma prova de que é homeomorfo a se (). \tablesmodenone
Capítulo \thechapter Introduction
History of the problem
Consider the set of all regular closed curves in the plane (i.e., immersions ), furnished with the topology (). The Whitney-Graustein theorem (, thm. 1) states that two such curves are homotopic through regular closed curves if and only if they have the same rotation number (where the latter is the number of full turns of the tangent vector to the curve).111Numbers enclosed in brackets refer to works listed in the bibliography at the end. Thus, the space has an infinite number of connected components , one for each rotation number . A typical element of () is a circle traversed times, with the direction depending on the sign of ; contains a figure eight curve.
For curves on the unit sphere , there is no natural notion of rotation number. Indeed, the corresponding space of immersions (i.e., regular closed curves on ) has only two connected components and ; this is an immediate consequence of a much more general result of S. Smale (, thm. A). The component contains all circles traversed an odd number of times, and the component contains all circles traversed an even number of times. Actually, the Hirsch-Smale theorem implies that , where denotes the set of all continuous closed curves on , with the compact-open topology; the properties of the latter space are well understood (see , §16).222The notation (resp. ) means that is homotopy equivalent (resp. homeomorphic) to .
In 1970, J. A. Little formulated and solved the following problem: Let denote the set of all closed curves on which have nonvanishing geodesic curvature, with the topology; what are the connected components of ? Although his motivation to investigate appears to have been purely geometric, this space arises naturally in the study of a certain class of linear ordinary differential equations (see  for a discussion of this class and further references).
Little was able to show (see , thm. 1) that has six connected components, , and , where the sign indicates the sign of the geodesic curvature of a curve in the corresponding component. A homeomorphism between and is obtained by reversing the orientation of the curves in .
The topology of the space has been investigated by quite a few other people since Little. We mention here only B. Khesin, B. Shapiro and M. Shapiro, who studied and similar spaces in the 1990’s (cf. , ,  and ). They showed that are homotopy equivalent to , and also determined the number of connected components of the spaces analogous to in , and , for arbitrary .
The first pieces of information about the homotopy and cohomology groups and for were, however, only obtained a decade later by N. C. Saldanha in  and . Finally, in the recent work , Saldanha gave a complete description of the homotopy type of and other closely related spaces of curves on . He proved in particular that
The reason for the appearance of an factor in all of these results is that (unlike Saldanha, cf. ) we have not chosen a basepoint for the unit tangent bundle ; a careful discussion of this is given in §1.
Overview of this work
The main purpose of this thesis is to generalize Little’s theorem to other spaces of closed curves on . Let be given and let be the set of all closed curves on whose geodesic curvatures are restricted to lie in the interval , furnished with the topology (for some ); in this notation, the spaces and discussed above become and , respectively. We present a direct characterization of the connected components of in terms of the pair and of the properties of curves in . It is shown in particular that the number of components is always finite, and a simple formula for it in terms of and is deduced.
More precisely, let , , where we adopt the convention that takes values in , with and . Also, let denote the greatest integer smaller than or equal to . Then has connected components , where
and contains circles traversed times (). The component also contains circles traversed times, and contains circles traversed times, for . In addition, it will be seen that each of is homotopy equivalent to ().
This result could be considered a first step towards the determination of the homotopy type of in terms of and . In this context, it is natural to ask whether the inclusion is a homotopy equivalence; as we have already mentioned, the topology of the latter space is well understood. It will be shown that the answer is negative when . We expect this to be false except when and . Actually, we conjecture that and have different homotopy types if and only if , but here it will only be proved that is homeomorphic to if ( and ).
Brief outline of the sections
It turns out that it is more convenient, but not essential, to work with curves which need not be . The curves that we consider possess continuously varying unit tangent vectors at all points, but their geodesic curvatures are defined only almost everywhere. This class of curves is described in §1, where we also relate the resulting spaces of curves to the more familiar spaces of curves. In this section we take the first steps toward the main theorem by proving that the topology of depends only on . A corollary of this result is that any space is homeomorphic to a space of type ; the latter class is usually more convenient to work with. Some variations of our definition are also investigated. In particular, in this section we consider spaces of non-closed curves.
In §2, we study curves which have image contained in a hemisphere. Almost all of this section is dedicated to proof that it is possible to assign to each such curve a distinguished hemisphere containing its image, in such a way that depends continuously on .
The main tools in the thesis are introduced in §3. Given a curve , we assign to certain maps and , called the regular and caustic bands spanned by , respectively. These are “fat” versions of the curve, and each of them carries in geometric form important information on the curve. We separate our curves into two main classes, called condensed and diffuse, depending on the properties of its caustic band. This distinction is essential throughout the work.
In §4, the grafting construction is explained. If the curve is diffuse, then we can use grafting to deform it into a circle traversed a certain number of times, which is the canonical curve in our spaces. We reach the same conclusion for condensed curves, using very different methods, in §5, where a notion of rotation numbers for curves of this type is also introduced. Although there exist curves which are neither condensed nor diffuse, any such curve is homotopic to a curve of one of these two types. The main results used to establish this are presented in .
In §7, we decide when it is possible to deform a circle traversed times into a circle traversed times in . It is seen that this is possible if and only if (where ), and an explicit homotopy when this is the case is presented. It is also shown that the set of condensed curves in with fixed rotation number is a connected component of this space.
The proofs of the main theorems are given in §8, after most of the work has been done. A direct characterization of the components of in terms of the properties of a curve is presented at the end of this section.
The last section is dedicated to the proof that the inclusion is not a (weak) homotopy equivalence if
Finally, we present in an appendix some basic results on convexity in that are used throughout the thesis. Although none of these results is new, complete proofs are given.
Capítulo \thechapter Spaces of Curves of Bounded Geodesic Curvature
Basic definitions and notation
Let denote either the euclidean space or the unit sphere , for some . By a curve in we mean a continuous map . A curve will be called regular when it has a continuous and nonvanishing derivative; in other words, a regular curve is a immersion of into . For simplicity, the interval where is defined will usually be .
Let be a regular curve and let denote the usual Euclidean norm. The arc-length parameter of is defined by
and is called the length of . Since for all , is an invertible function, and we may parametrize by . Derivatives with respect to and will be systematically denoted by a and a , respectively; this convention extends, of course, to higher-order derivatives as well.
Up to homotopy, we can always assume that a family of curves is parametrized proportionally to arc-length.
Let be a topological space and let be a continuous map from to the set of all regular curves () with the topology. Then there exists a homotopy , , such that for any :
and is parametrized so that is independent of .
is an orientation-preserving reparametrization of , for all .
Let be the arc-length parameter of , its length and the inverse function of . Define by:
Then is the desired homotopy. ∎
The unit tangent vector to at will always be denoted by . Set for the rest of this section, and define the unit normal vector to by
where denotes the vector product in . Equivalently, is the unique vector which makes a positively oriented orthonormal basis of .
Assume now that has a second derivative. By definition, the geodesic curvature at is given by
Note that the geodesic curvature is not altered by an orientation-preserving reparametrization of the curve, but its sign is changed if we use an orientation-reversing reparametrization. Since the sectional curvatures of the sphere are all equal to 1, the normal curvature of is 1 at each point. In particular, its Euclidean curvature ,
Closely related to the geodesic curvature of a curve is the radius of curvature of at , which we define as the unique number in satisfying
Note that the sign of is equal to the sign of .
A parallel circle of colatitude , for , has geodesic curvature (the sign depends on the orientation), and radius of curvature or at each point. (Recall that the colatitude of a point measures its distance from the north pole along .) The radius of curvature of an arbitrary curve gives the size of the radius of the osculating circle to at , measured along and taking the orientation of into account.
If we consider as a curve in , then its “usual” radius of curvature is defined by . We will rarely mention or again, preferring instead to work with and , which are their natural intrinsic analogues in the sphere.
Spaces of curves
Given and of norm 1, there exists a unique having as first column and as second column. We obtain thus a diffeomorphism between and the unit tangent bundle of .
For a regular curve , its frame is the map given by
In other words, is the curve in associated with , under the identification of with . We emphasize that it is not necessary that have a second derivative for to be defined.
Now let and . We would like to study the space of all regular curves satisfying:
for each .
Here is the identity matrix and is the geodesic curvature of . Condition (i) says that starts at in the direction and ends at in the direction .
This definition is incomplete because we have not described the topology of , nor explained what is meant by the geodesic curvature of a regular curve (which need not have a second derivative, according to our definition). The most natural choice would be to require that the curves in this space be of class , and to give it the topology. The foremost reason why we will not follow this course is that we would like to be able to perform some constructions which yield curves that are not . For instance, we may wish to construct a curve of positive geodesic curvature by concatenating two arcs of circles and of different radii, as in fig. 3 below. Even though the resulting curve is regular, it is not possible to assign any meaningful value to the curvature of at . However, we may approximate as well as we like by a smooth curve which does have everywhere positive geodesic curvature. We shall adopt a more complicated definition precisely in order to avoid using convolutions or other tools all the time to smoothen such a curve.
A function is said to be of class if it is an indefinite integral of some . We extend this definition to maps by saying that is of class if and only if each of its component functions is of class .
Since , an function is absolutely continuous (and differentiable almost everywhere).
We shall now present an explicit description of a topology on which turns it into a Hilbert manifold. The definition is unfortunately not very natural. However, we shall prove the following two results relating this space to more familiar concepts: First, for any , , the subset of consisting of curves will be shown to be dense in . Second, we will see that the space of regular curves satisfying conditions (i) and (ii) above, with the topology, is (weakly) homotopy equivalent to .444The definitions given here are straightforward adaptations of the ones in , where they are used to study spaces of locally convex curves in (which correspond to the spaces when ).
Consider first a smooth regular curve . From the definition of we deduce that
is called the logarithmic derivative of and is the geodesic curvature of .
Conversely, given and a smooth map of the form
let be the unique solution to the initial value problem
Define to be the smooth curve given by . Then is regular if and only if for all , and it satisfies if and only if for all . (If for all then is regular, but is obtained from by changing the sign of the entries in the second and third columns.)
For each pair , let be the smooth diffeomorphism
and, similarly, set
Let satisfy . A curve will be called -admissible if there exist and a pair such that for all , where is the unique solution to equation (4), with given by
When it is not important to keep track of the bounds , we shall say more simply that is admissible.
In vague but more suggestive language, an admissible curve is essentially an frame such that has geodesic curvature in the interval . The unit tangent (resp. normal) vector (resp. ) of is thus defined everywhere on , and it is absolutely continuous as a function of . The curve itself is, like , of class . However, the coordinates of its velocity vector lie in , so the latter is only defined almost everywhere. The geodesic curvature of , which is also defined a.e., is given by
The reason for the choice of the specific diffeomorphism in (5) (instead of, say, ) is that we need to diverge linearly to as in order to guarantee that whenever . The reason for the choice of the other diffeomorphisms is analogous.
Let , . Define to be the set of all -admissible curves such that
where is the frame of . This set is identified with via the correspondence , and this defines a (trivial) Hilbert manifold structure on .
In particular, this space is contractible by definition. We are now ready to define the spaces , which constitute the main object of study of this work.
Let , . We define to be the subspace of consisting of all curves in the latter space satisfying
Here is the frame of and is the identity matrix.555The letter ‘L’ in is a reference to John A. Little, who determined the connected components of in .
Because has dimension 3, the condition implies that is a closed submanifold of codimension 3 in . (Here we are using the fact that the map which sends the pair to is a submersion; a proof of this when and can be found in 3 of , and the proof of the general case is analogous.) The space consists of closed curves only when . Also, when and simultaneously, no restrictions are placed on the geodesic curvature. The resulting space (for arbitrary ) is known to be homotopy equivalent to ; see the discussion after (\thechapter.13).
Note that we have natural inclusions whenever . More explicitly, this map is given by:
it is easy to check that the actual curve associated with the pair of functions in on the right side (via (3), (4) and (6)) is the original curve , so that the use of the term “inclusion” is justified. In fact, this map is an embedding, so that can be considered a subspace of when .
The next lemma contains all results on Hilbert manifolds that we shall use.
Let be a Hilbert manifold. Then:
is locally path-connected. In particular, its connected components and path components coincide.
If is weakly contractible then it is contractible.666Recall that a map between topological spaces and is said to be a weak homotopy equivalence if is an isomorphism for any and . The space is said to be weakly contractible if it is weakly homotopy equivalent to a point, that is, if all of its homotopy groups are trivial.
Assume that is a regular value of . Then is a closed submanifold which has codimension and trivial normal bundle in .
Let and be separable Banach spaces. Suppose is a bounded, injective linear map with dense image and is a smooth closed submanifold of finite codimension. Then is a smooth closed submanifold of and is a homotopy equivalence of pairs.
Part (a) is obvious and part (b) is a special case of thm. 15 in . The first assertion of part (c) is a consequence of the implicit function theorem (for Banach spaces). The triviality of the normal bundle can be proved as follows: Let and be the fiber over of the normal bundle . Then
and lies in the kernel of the derivative by hypothesis, as vanishes identically on . Since is surjective and , must be an isomorphism when restricted to . This is valid for any , so we can obtain a trivialization of by setting:
Finally, part (d) is thm. 2 in . ∎
Let . Then the subset of all of class is dense in .
This follows from the fact that the set of smooth functions is dense in . ∎
Let , and , . Define to be the set, furnished with the topology, of all regular curves such that:
for each .
The value of is not important, as all of these spaces are homotopy equivalent. Because of this, after the next lemma, when we speak of , we will implicitly assume that .
Let (), and . Then the set inclusion is a homotopy equivalence.
In this proof we will highlight the differentiability class by denoting by . Let , let (where denotes the set of all functions , with the norm) and let be set inclusion. Setting , we conclude from (\thechapter.7(d)) that is a homotopy equivalence. We claim that , where the homeomorphism is obtained by associating a pair to the curve obtained by solving (4) (with defined by (3) and (6) and ), and vice-versa.
Suppose first that . Then (resp. ) is a function of class (resp. ). Hence, so are and , since and are smooth. Conversely, if , then is of class and of class , and the frame of the curve corresponding to that pair satisfies
Since the entries of are of class (at least) , the entries of are functions of class . Moreover, , hence
and the velocity vector of is seen to be of class . It follows that is a curve of class . Finally, it is easy to check that the correspondence is continuous in both directions. ∎
The (two-sheeted) universal covering space of is . Let us briefly recall the definition of the covering map .777See  for more details and further information on quaternions and rotations. We start by identifying with the algebra of quaternions, and with the subgroup of unit quaternions. Given , , define a transformation by . One checks easily that preserves the sum, multiplication and conjugation operations. It follows that, for any ,
where denotes the usual inner product in . Thus is an orthogonal linear transformation of . Moreover, (where is the unit of ), hence the three-dimensional vector subspace consisting of the purely imaginary quaternions is invariant under . The element is the restriction of to this subspace, where is identified with the quaternion .
In what follows we adopt the convention that (resp. ) is furnished with the Riemannian metric inherited from (resp. ).
Let denote the metric in and the metric in . Then , where denotes the pull-back of by .
It suffices to prove that if
is a regular curve and then . Let us assume first that , so that . From the definition of , we have
and similarly for , , where, as above, we identify with the imaginary quaternions. Hence