Abstract
We present some fine properties of immersions between manifolds, with particular attention to the case of immersed curves . We present new results, as well as known results but with quantitative statements (that may be useful in numerical applications) regarding tubular coordinates, neighborhoods of immersed and freely immersed curve, and local unique representations of nearby such curves, possibly “up to reparameterization.” We present examples and counterexamples to support the significance of these results. Eventually, we provide a complete and detailed proof of a result first stated in a 1991-paper by Cervera, Mascaró, and Michor: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold.
1. Introduction
In general, let and be smooth finite dimensional connected Hausdorff paracompact manifolds without boundary, with .
This paper studies properties of immersions that are maps such that is full rank at each .
A particular but very interesting case are closed immersed curves that are maps with at all , where be the circle in the plane. They will be called planar when .
This paper is mostly devoted to this case (a forthcoming paper [15] will generalize many results in this paper to the general case of immersions ).
Immersed planar curves have been used in computer vision for decades; indeed, the boundary of an object in an image can be modeled as a closed embedded curve, by the Jordan Theorem. Possibly, the first occurrence was active contours, introduced by [9] and used for the segmentation problem: the idea is to minimize energy, defined on contours or curves, that contains an image-based edge attraction term and a smoothness term, which becomes large when the curve is irregular. An evolution is derived to minimize the energy based on principles from the calculus of variations. There have been many variations to original model of [9]; for example [6] and a survey in [3].
An unjustified feature of the model of [9] was that the evolution is dependent on the way the contour is parameterized. Thereafter, the authors of [4, 11] considered minimizing a geometric energy, which is a generalization of Euclidean arclength, defined on curves for the edge-detection problem. The authors derived the gradient descent flow in order to minimize the geometric energy.
This lead to a principle: all operations related to curves should be independent of the choice of parameterizations.
Operations on the space of curves are best described and studied if the whole space of curves is endowed with a differential structure, so that it becomes a smooth manifold.
The above two remarks lead to the following question. If is the space of curves that we are interested in and is the action of reparameterization, then the quotientis the space of curves up to parameterization (also called geometric curves in the following): when (and how) can we say that this quotient is a smooth manifold?
This was discussed in [16], using a result from [5].
A purpose of this paper is to revisit the key result in [5]: indeed the proof in that paper is missing two key steps.
1.1. Plan of the Paper
In Section 2, we will define the needed topologies on the space of functions; we will present well known definitions and notations for curves, such as derivation and integration in arc parameter, length, normal vectors, and curvature; we will classify immersed and freely immersed curves and present results and examples.
In Section 3, we will present advanced results for immersed curves; we will discuss representation of nearby curves in tubular coordinates; we will show how the open neighborhood of a curve in the space of curves can be defined using tubular coordinates, so that if is immersed (respectively, freely immersed) then all curves in the neighborhood are immersed (respectively, freely immersed); we will show with examples what goes wrong when hypotheses are not met.
In Section 4, we will present the proof of this theorem: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold. We will then explain, in a step by step analysis, why the original proof in [5] was incorrect.
A supplemental file contains Wolfram Mathematica code to generate some of the figures.
2. Definitions
In this section, we will present well-known definitions and results regarding immersions, with particular attention to immersed curves.
2.1. Topologies
Definition 1. We denote by the space of all maps that are of class . Here, .
There are classically two types of topologies for this space.(i)The weak topology, as defined in Ch. 1 Sec. 1 in [8], that coincides with the compact-open -topology as defined in 41.9 in [12]; if , then the “weak topology” is the topology of the Fréchet space of local uniform convergence of functions and their derivatives up to order (ii)The strong topology as defined Ch. 1 Sec. 1 in [8] coincides with the Whitney -topology as defined in 41.10 in [12]
If is compact, then the two above coincide; if moreover and , then is the usual Banach space.
Remark 1. If but is not compact, then “strong topology” does not make a topological vector space since it has uncountably many connected components; but the connected component containing contains only compactly supported functions, and it has the topology (as defined in 6.9 in Rudin [18]) that is the strict inductive limit (for the definition of strict inductive limit and its properties, we refer to 17G at page 148 in [10]) of the immersionswhere for each compact, is the space of that are zero outside of , with a standard Banach (or Fréchet, for ) structure.
Proposition 1. The sets of immersions, submersions, and embeddings are open in with the strong topology, for .
Proofs are in Ch. 1 Sec. 1 in [8].
Definition 2. For , let be the family of diffeomorphisms of : all the maps that are and invertible, and the inverse is . It is a group, the group operation being “composition of functions.”
Proposition 2. is open in with the strong topology.
See Thm. 1.7 in Ch. 1 Sec. 1 in [8].
We will omit the superscript “” from in the following, for ease of notation.
2.2. Immersions
2.2.1. Free Immersion
Definition 3. An immersion is called “free” if for implies that is the identity.
Proposition 3 (see [5] Lemma 1.3). If is immersed and for all and for a , then .
Proof. Indeed, it is easily seen thatis closed; and it is also open, since an immersion is also a local diffeomorphism with its image.
As a corollary, if and and for a , then . Another corollary states the following.
Corollary 1 (see [5] Lemma 1.4). If is an immersion and there is a s.t. for one and only one , then is a free immersion.
This implies that, when , the free immersions are a dense subset of all immersions (for all the topologies considered in this paper).
2.2.2. Reparameterizations and Isotropy Group
We first consider the general case of immersions .
Definition 4. The isotropy group (a.k.a. “stabilizer subgroup” or “little group”) is the set of all such that ; it is a subgroup of .
Obviously, is freely immersed if and only if contains only the identity.
We will prove that is discrete and finite when is compact.
Remark 2. If we reparameterize , then changes by conjugation:
Remark 3. If is orientable, then has a subgroup of orientation preserving diffeomorphisms; for the case of curves, then we obtain that has two connected componentswhere(i)is the family of diffeomorphisms with , and is a normal subgroup(ii)is the family of diffeomorphisms with
Consider a curve and let be its isotropy group: we will prove in Lemma 3 that if , then .
We will mostly use in the following.
Note that is a perfect group [19] (see [13] for a self-contained presentation); it is also a simple group: see Discussion in Sec. 2 in [2] for further references. (The author thanks Prof. Kathryn Mann for her help on these subjects.)
2.3. Curves
Remember that is the circle in the plane. We will often associate , for convenience. In this case, we will associate .
Definition 5. A closed curve is a map . We will always assume that the curve is of class (at least). The image of the curve, or trace of the curve, is .
When convenient, we will (equivalently) view as (that is, modulus translations), and consequently a closed curve will be a map that is -periodic.
In particular, this will be the correct interpretation when we will write the operation for .
Remark 4. The “distance” of points in will be the intrinsic distance; this distance will be represented by the notation:for , and it is the length of the shortest arc in connecting the two points . Note that if we identify to and pick two points and represent them as real numbers, it may happen that
Definition 6. (basepoint). We will select a distinguished point in the circle : for , it will be ; for , it will be ; for , it will be .
Given a curve as above, we will call the basepoint for the curve.
Example 1 (of a nonfreely immersed curve). The doubly traversed circle defined as(i) for when we consider , or equivalently(ii) for that we identify with Setting , we have that , so is not freely immersed.
Example 2 (taken from [5]). Note that there are free immersions without a point with only one preimage: consider a “figure eight” which consists of two touching circles. Now, we may map the circle to the figure eight by going first three times around the upper circle, then twice around the lower one. This immersion is free.
We provide a simple Example 3 that shows how such curve can be made smooth.
2.3.1. Length, Tangent, and Curvatures
In the following, let be an immersed curve.
Definition 7. If the curve is immersed, we can define the derivation with respect to the arc parameter
We will write instead of when we are dealing with multiple curves, and we will want to specify which curve is used.
Definition 8. We define the tangent vector
Definition 9. The length of the curve is
Definition 10. We define the integration by arc-parameter of a function along the curve by
There are two different definitions of curvature of an immersed curve: mean curvature and signed curvature , which is defined when is valued in .
and are extrinsic curvatures, they are properties of the embedding of into .
Definition 11 (H). If is regular and immersed, we can define the (mean) curvature of as
It is easy to prove that .
Definition 12 (N). When the curve is planar, we can define a normal vector to the curve, by requiring that , and is rotated degree anticlockwise with respect to .
Definition 13. () If is in and , then we can define a signed scalar curvature , so that
There is a choice of sign in the above two definitions; this choice agrees with the choice in [21].
When we will be dealing with multiple curves, we will specify the curve as a subscript, e.g., will be the tangent, curvature, and normal to the curve .
Remark 5. Note that are geometrical quantities. If and , then , , and .
2.3.2. Arc Parameter
Let be an immersed planar curve. We recall this important transformation.
Lemma 1 (constant speed reparameterization). A curve can be reparameterized to using a so that where is constant.
Proof. For simplicity, we assume that . Let , let . Then is a diffeomorphism, let .
Reparameterization to constant speed is a smooth operation in the space of curves, see Theorem 7 in [20].
When , we will say that the curve is by arc parameter. A curve can be reparameterized to arc parameter without changing its domain (as done above) iff (if this is not the case, we will rescale the curve to make it so.)
2.3.3. Angle Function and Rotation Index
Proposition 4 (angle function and rotation index). If is planar and is immersed, then is continuous and , so there exists a continuous function satisfyingand is unique, up to adding the constant with . is called the angle function.
Moreover , where is an integer, known as rotation index of . This number is unaltered if is deformed by a smooth homotopy (Figure 1).
See 2.1.4 in [1] or Theorem 53.1 in [17], and following.
Remark 6. We can use the angle function to compute the scalar curvature , that was defined in Definition 13 by , indeed deriving (14) and combining this withwe obtain
2.4. Shapes
Shapes are usually considered to be geometric objects. Representing a curve using forces a choice of parameterization that is not really part of the concept of “shape.”
Suppose that is a space of immersed curves .
Definition 14 (geometric curves). The quotient space is the space of curves up to reparameterization, also called geometric curves in the following. Two parametric curves such that for a are the same geometric curve inside .
is mathematically defined as the set of all equivalence classes of curves that are equal but for reparameterization,
We may also consider the quotient w.r.t . The quotient space is the space of geometric-oriented curves.
Unfortunately, the quotient of immersed curves by reparameterizations is not a manifold; but the quotient of freely immersed curve is.
Theorem 1. Suppose that is the space of the freely immersed curves; and that and have the topology of the Fréchet space of functions, then the quotient is a smooth manifold modeled on .
One aim of this paper will be to give a complete proof of this result, first presented in [5]. (We remark that the theorem in [5] was presented for the case of immersions ). The proof is in Section 4.1. Indeed, as we will discuss in Section 4.2, the proof in [5] misses some key arguments.
3. Advanced Properties of Immersed Curves
In this section, we will present results regarding immersed curves that are either new or presented in more precise form than usually found in the literature.
Most of the results are presented, for sake of simplicity, for planar curves , but can be extended to the case of curves taking values in a manifold , up to some nuisance in notations.
The general case of immersions requires instead some arguments that will be discussed in a future paper [15].
3.1. Examples
Definition 15. We start with some classical examples of functions of compact support. Let(see Figure 2 on the following page).
We will use these to build some following examples.
Example 3. We present here a simple smooth formula for Example 2this is a function depicted at Figure 3 on the next page.
3.1.1. Trace and Parameterization
If a curve is embedded then the curve is identified by its image, in these senses.(i)If are embedded and have the same image, then there is an unique reparameterization such that .(ii)Suppose that is embedded and is the trace; suppose that is parameterized by constant speed parameter; let us fix a candidate basepoint in the trace. We can state that characterize the embedded curve up to a choice of direction: precisely, there are exactly two different , parameterized by constant speed parameter, such that , and they satisfy
for unique choices of (dependant on ).
In particular, if the rotation index of is , then the latter curves have rotation indexes .
Since the definition of freely immersed curve says that the curve identifies a unique parameterization, then we may be induced to think that the above two properties extend to freely immersed curves: but this is not the case.
Example 4. The following two curves have the same trace, are freely immersed, are smooth, but have rotation indexes 0 and 1.(1)This immersed closed curve with components(2)This immersed closed curve with componentsSee Figure 4 on the following page.
3.2. Reparameterizations and Isotropy Group
Lemma 2. If , then has two fixed points.
Proof. We represent as a map that is continuous, strictly decreasing and such thatthenso the graph must intersect both the graph and the graph for two different points that are the two fixed points.
Lemma 3. If is immersed and , then .
Proof. Suppose that , let be a fixed point (by Lemma 2). By derivingsetting and this is impossible since .
3.3. Local Embedding
3.3.1. Length of Curve Arcs
Definition 16. Suppose is . Let . When there are two arcs in connecting to . Bywe will mean the minimum of the lengths of when restricted to one of the two arcs connecting to .
If is periodically extended to and , then there is an unique such thatand then, lettingwe define
In particular, when is parameterized at constant speed (i.e., ), then we will (covertly) assume that are chosen (up to adding ) so that and then
Remark 7. When is not parameterized by constant velocity, the above may lead to some confusion. The interval in the notation (28) implicitly refers to the choice of arc in that provides the above minimum. Note that this may not be the shortest arc connecting to in . This may happen if the parameterization of has regions of fast and slow velocity, as in this example.
Example 5. Let be the standard circle, and(see plot in Figure 5) then smooth out the corners of so that it becomes a diffeomorphism of ; letlet in , then , and is given by the arc moving counterclockwise from to , whileis given by the arc moving clockwise from to .
This never happens for small distances/lengths, though.
Theorem 2. Fix an immersed curve ; let(i)For any in such that the shortest arc connecting them in is also the arc where is computed(ii)For anyinsuch that the arc where is computed is also the shortest arc connecting them in , whose length is(iii)In any of the above cases,
3.3.2. Estimates
We begin with this estimate.
Proposition 5. Let be the angle function, for . The fact that the curve is closed imposes lower bounds on .(i)If the rotation index of the curve is not zero, then so necessarily (ii)If the rotation index of the curve is zero, then necessarily Indeed, we can prove thatotherwise, lettaking , we would havefor all , hence the curve would not be closed.
Definition 17. Given , a immersed closed curve, we recall that is the scalar curvature of ; we define Note that since the curve is closed, cannot be identically zero.
Note that , but we define two quantities since this simplifies the notation in the following. We have but .
Remark 8. Note that if we rescale the curve by a factor , then and are multiplied by as well. If we rotate or translate , then and are unaffected. If we reparameterize, then are unchanged, whereas if and , we haveIn all (with the exception of relation (100) in Lemma 7) following definitions, propositions, and theorems, the formulas are built to be “geometrical”: this means that, if the curves are reparameterized, rescaled, translated or rotated, then the formulas change in predictable ways (as explained above).
This simplifies the proofs: in the proofs we can assume, with no loss of generality, that the curve is parameterized by arc parameter.
Remark 9. Note that for curves of index zero and for curves of index .
Proof. We use Proposition 5. The formula in the thesis is invariant for reparameterizations and scaling; we rescale the curve so that and reparameterize by arc parameter so that . For curves of index zero, the thesis , that is, ; since by (16), this last becomes that was proved above. For curves of index , the thesis , that is, , then becomes that was proved above.
For the above is sharp, as in the case of .
3.3.3. Local Embedding of Curves
It is well known that a immersion is a local embedding. For curves of class , we can provide a simple quantitative statement.
Proposition 6 (local embedding). Let a immersed curve. Define as in Definition 17. For any , let ; assume that , then ; so, is embedded.
Proof. For simplicity, we assume that is periodically extended to ; then, we identify the interval in that is associated to the arc of the curve where the length is computed; for simplicity, we call this interval again. (if the arc is short enough, then by Theorem 2, no ambiguity is possible).
Using Lemma 1 and Remark 8, assume that ; then, , so andAs noted in (16)so . Let be the middle point. Let be the angle function (14). Up to rotation, suppose ; so, we can assume . Let . For any , we have ; hence, ; hence, for all , we haveand hence, ; hence, for , for the abscissa, we can write
3.4. Isotropy Group Is Discrete
Given an immersion , it is possible to prove that the isotropy group is discrete (when is paracompact) and even finite (when is compact; this latter result appears in [5]). When considering curves, we can obtain the same results (and even more) in a more direct and geometric way.
Lemma 4. Let be immersed.(i)is finite(ii)If and for an , then (iii)If is parameterized by constant speed (see Lemma 1), then there is a s.t. is the set of all for (the proof of this is a special case of the 2nd step of the proof of Lemma 9)
Proof. (i)We prove the third point. Indeed deriving and noting that , we obtain so ; hence, , for all ; if is irrational, then would be dense in , and this is denied by Proposition 6. Moreover, if and , then ; but by Proposition 6, , that is, . So, there is an unique such that any can be written as .(ii)The above characterization shows that if and , then . By Remark 2, this is valid for any curve (even when it is not parameterized by constant speed). This proves the second point.(iii)The first point follows again from Remark 2.
3.5. Tubular Neighborhoods
Existence of tubular neighborhood is well known; we provide a quantitative result for planar immersed curves.
Proposition 7 (tubular neighborhood). Define as in Definition 17. Fix with . Letthen is a diffeomorphism with its image. Moreover, if the arc is contained in the arc identified above, thenwhereas (obviously)
Proof. Assume that the curve has length and is parameterized in arc parameter; with no loss of generality (recalling Remark 8); let be the angle function (14).
Extend to a periodic function and identify the interval in that is associated with the arc of the curve where the length is computed. For simplicity, we call this interval again (if the arc is short enough, then by Theorem 2, no ambiguity is possible).
The Jacobian of isso its determinant is by the hypothesis .
We will then prove that is injective, so it will be an homeomorphism with its image, and since it is a local diffeomorphism, it will be a diffeomorphism.
Choose and with and , .
We set . Up to rotation, we assume that and , so that is perpendicular to the axis. As in Proposition 6, we can prove that for all .
We writeand then for the abscissanote that . Deriving, we obtainWe then obtain thatwhileand recalling that and summing, we obtain
We will call tubular coordinates around the formula (52).
3.5.1. Counterexample
The hypothesis “” in the previous proposition may be broadened to “”; but the results fails if we only assume that “” with , as seen in this example (adapted from [1]).
Example 6. Let and ; then, for ,and this meets the axes forSo, by symmetry,and at the same time,
3.5.2. Nearby Points
Suppose is a immersed curve. Let and andand .
Proposition 8. is also the set of points at distance at most from the trace .
Proof. Let be the trace of the curve (it is a compact subset of ). We use the distance function defined as(for an introduction to this object, see [14] and references therein).
Let , there is a such thatSo,then let be a minimum forSo, clearly,Vice versa if let be a minimum as above, then geometrical considerations tell that the segment from to is orthogonal to the tangent at .
As a corollary of Proposition 7, for any such neighborhood of the image of , the “projection to ” is a multivalued map (with finitely many projections in ).
3.6. Not a Covering Map
By looking at the previous Proposition 7, we may think that is the universal covering map of (see [17] for the definition). This would be very convenient, and indeed, we could use the lifting lemma to ease some of the following proofs.
Suppose are immersed curves. Consider this statement, that is usually called lifting lemma:
If the trace of is contained in , then there is a choice of continuous , such that
Unfortunately, this is not the case, as seen in this example in Figure 6, where the curve is blue and the curve is red. The trace of the curve is all contained in the open set , but representation (72) cannot hold. We can though prove a version of the lifting lemma useful in the following.
3.7. Neighborhoods
3.7.1. Nearby Projection
Lemma 5 (nearby projection). Fix a immersed curve , with .(1)If and and then there is an with and a with such that Note also that is uniquely identified by .(2)They are unique in the family of such that and so we can see as functions of , as follows.(3)Consider and for whichlet small such that and let for convenience. There is a choice of function of class such thatfor all , and they are unique as specified above.
Note that .
Proof. Suppose is by arc parameter (with no loss of generality as explained in Remark 8); so we write (recalling (32)) instead of .(i)Choose as in the statement and let , then consider any minimum point of note that the minimum value has to be less than ; so, but at the same time (since and are at arc distance at most ), by the previous Proposition 6. So, combining the two but so is not at extremes. Then, any providing the minimum must be internal in the interval : by geometrical reasoning the segment from to is orthogonal to the curve so there is a such that(ii)Recall that ; the map is injective for and , so are unique.(iii)For any , we haveand since , then there is an unique and with such thatand we denote them by . Moreover, we can invert the functionand writefor . This proves that .
3.7.2. Global Lifting
Proposition 9 (global lifting). Suppose is a immersed curve and is , with . Fix . Suppose that we have for all . There exists choice of and such thatwith andholding for all . And, they are unique in the class of functions such that and
Note also that is uniquely identified by .
Proof. We just substitute in the previous Lemma. By the second point, we can define functions uniquely as prescribed. By the third point, they are .
Remark 10. Suppose we are given two curves and we know that there exists a choice such thatas in (88). If we rotate or translate the two curves, then the above relation will hold, with the same . If we rescale the two curves by , then the relation will hold with .
If we choose and we reparameterize all curves at the same time by , thenholds forThis follows from direct computation and Remark 5.
Example 7. So far so good, but may fail to be a diffeomorphism, as in this simple example in Figure 7. where the curve is blue and the curve is red.
But some simple lemmas can help.
Lemma 6. Let . If we have such thatthen the angle between and satisfies , and moreover,See Figure 8 on the next page.
Remark 11. Let . Suppose are maps andif and we reparameterize all curves at the same time by , thenSimilarly, if we rescale, rotate, or translate all curves at the same time.
So, (96) is a geometric estimate, indeed we may rewrite it aswhere .
Lemma 7. Assume all hypotheses in Proposition 9. Assume moreover that and assume thatthen is immersed and is a diffeomorphism. Moreover, when is parameterized at constant speed, we can state that
Proof. We rescale and reparameterize to arc parameter using a reparameterization , and at the same time, we rescale and reparameterize using the same rescaling and (note that is not necessarily by arc parameter); with no loss in generality, as explained in Remarks 10 and 11.
Let be the angle between and : by Lemma 6, , so it is at most . Let ; we know that ; the angle between and is at most so : so the angle between and is at most , and this is less than .
Deriving in and assuming that is by arc parameterwhere are evaluated at ; then,now if then ; moreover, by the above reasoning , so . Moreover, we note that , and to prove (100). Relation (93) tells then that will always be a diffeomorphism, for any curve satisfying the hypotheses.
We summarize all the above: we show sufficient hypothesis such that may be represented in tubular coordinates around .
Theorem 3 (representation theorem). Suppose is a immersed curve and is , with . Define as in Definition 17. Fix . Suppose that we have andfor all .
Then, is immersed, there are and of class such thatwith andholding for all .
They are unique in the class of functions such that and
Note also that is uniquely identified by .
Proof. We can rescale and reparameterize to arc parameter, and we rescale and reparameterize at the same time ( will not be by arc parameter in general); as discussed in Remarks 8, 10, and 11, the hypotheses and theses are unaffected by this action. Then we apply all previous results. Just note thatfor any diffeomorphism.
Remark 12. Actually, rerunning on the above proofs with some patience, we can improve the above thesis a bit. We add to the previous theorem these hypotheses: fix and then and suppose that we have andfor all .
Then, all above thesis hold, moreover, there are two continuous functions (independent on ) with , such that
3.7.3. Asymmetry
Warning: the previous theorem seems symmetric, but it is not. The caveat is in the constants : it may be the case that they are quite different from . In Figure 9, we see a piece of the two curves: curve is blue and is flat; curve is red, and it has two inflections points where the tangents are at an angle which is as small as we would like, but then the inflection points can also be so close that the normals will cross before reaching curve . So, while there is an easy way of representing using tubular coordinates around , there is no way to find so as to write
3.7.4. Vice Versa
We have also a sort of vice versa of the previous Theorem 3.
Proposition 10 (derepresentation). Suppose is a immersed curve and are given by tubular coordinateswhere and are of class ; define(where is the curvature of ). Then,and (obviously) , for all .
Proof. We rescale and reparameterize by arc parameter, and we rescale and reparameterize along with , as explained in Remark 11; this operation is justified by Remarks 8 and 10; in particular, note that are scale invariant; then,soHence,
3.7.5. Loss of Regularity
Unfortunately, the representation discussed above suffers from a loss of regularity. Indeed, if is and is , it may be the case thatis of class but not of class .
This can be seen in very simple examples.
Example 8. Suppose that, for near , we havesuch a curve is but not ; then, for ,(that can be easily computed using a standard formula for curvature of planar curves, see Sec. 1.7.1 in [21]).
Choose then , sobut thenand for ,So,but for .
3.7.6. Neighborhoods of a Curve
The above results encode two different but equivalent ways to define a topology on the “manifold of immersed curves.” We specify them by describing the local bases of neighborhoods of a curve .(i)The “Banach way” in which a local base of open neighborhoods of a curve is given by the sets where is small, and(ii)The “geometric way” in which a neighborhood in the local base is defined, for small, as the set of all that can be expressed as in (88), namely, for all choices of and with wheremoreover derivatives of may be computed in arc-parameter.
The above are “equivalent” in this sense. Assume that the curve is .(i)For any that defines neighborhood of the first type, there is small enough that defines a neighborhood of the second type, so that ; this is easily proved (by using Leibnitz and Faa di Bruno formulas).(ii)Consider now a neighborhood of the second type, for an small; for small enough, the previous results Theorem 3 and Remark 12 tells us that any curve can be expressed in tubular coordinates; since tubular coordinates are a local diffeomorphism, similar arguments as above (plus Theorem 2) show that (for even smaller) .
We skip details for sake of brevity.
In all the above there is, however, an annoying condition: to prove equivalence of neighborhoods, we have to assume that the curve is . For this reason, this works well for defining topologies in the “manifold of smooth immersed curves”; in this case, we will use neighborhoods of the first kind (or, respectively, of the second kind) for all and all . This is the common approach, see [12].
3.7.7. Local Injectivity
So far, we have considered parametric curves. We have seen in Theorem 3 that we can represent nearby curves in a unique way using tubular coordinates, i.e., the map .
What happens when we consider geometric curves, that is, curves up to parameterization?
Lemma 8 (local injectivity). Let be a freely immersed planar curve. There exists a such that, ifwithwhere is the arc derivation: then and is the identity.
Proof. If rescale the curves and the functions by a constant , and we rescale the constant by the same constant , then all hypotheses and theses are unaffected. So, we can assume with no loss of generality that has length .
If we reparameterize then (cf the relation (93) in Remark 10) the functions are reparameterized as well; having , then ; and similarly for , again hypotheses and theses are unaffected.
So, we can assume that is parameterized by arc parameter with no loss of generality.
Suppose thatwith and , thenSo,Suppose moreover,with , thenand thenBy contradiction, we may writewithwhere is not the identity: then, when , the uniqueness condition (90) is contradicted, so there is a such thatSo, using Theorem 2,Up to a subsequence, we can assume that andWe know thatup to a subsequence uniformly and uniformly, where is a bi-Lip homeomorphism. Then, uniformly, and passing to the limitso is a diffeomorphism. Moreover,so cannot be the identity; hence, is not freely immersed.
3.7.8. Auto Representation (This Section May Be Skipped on a First Read, Since It Is Not Needed in the following)
The above result is very important, but the proof gives no hint on what is going on. To this end, we drop the requirement that the curve be freely immersed and look at an easy question. Is it possible for a curve to represent itself locally?
Lemma 9. Let be a immersed planar curve. There are only finitely many ways in which the curve can represent itself geometrically, that is, finitely many choices with and continuous with . In particular, there is a such that, if then .
In particular, if is freely immersed, then implies and .
Proof. Define as in Definition 17 (see also Proposition 7). Let . Consider the family of all the pairs with and andwe will prove that there are only finitely many such pairs.
Hence, we will let be smaller than the minimum of for all such pairs:In the example, in Figure 10, there are 3 pairs in .
We have some very strong properties.(i)If and there is a s.t., then and . Indeed, there is a small interval containing , where we can invert the map and that is, the first component of is , so in ; so, the set is both open and closed. The previous argument also proves that .(ii)Figure 11 on the following page can be used as a visual guide in the following proof. Let be an open interval such that the length of is less than and more than . As noted in Remark 9, . Let be the image of for and . Choose with , let we will prove that either , or , , and . Assume that , let , then ; so, using the relation (146), and the fact that is a diffeomorphism for and , we obtain that so ; hence, by the previous point , .(iii)The differential of (computed using arc-length derivative) isso the smallest principal value is at least . We can estimate the length of for to be at least . Hence, there can be only finitely many such intervals.
Example 9. The constants in Lemma 9 and in Lemma 8 though cannot be estimated apriori by using differential quantities such as . These constants may arbitrarily smaller than the quantity (defined in Definition 17) that provides the width of the tubular neighborhood (Proposition 7). They really depend on how the curve is drawn.
This is seen in simple examples as follows:(i) for , or equivalently for that are small perturbations of the doubly traversed circle seen in Example 1; see Figure 12. This curve is freely immersed, but it is quite near to the doubly traversed circle that is not freely immersed. For small, the curvature of is approximatively 1; so, , andwhileHence,so we can use Theorem 3 to expressusing tubular coordinates when , with : so ; similarly, we can represent using , so we have .
3.8. Free Immersions Are Open
We have thus come to a fundamental result.
Theorem 4. Free immersions are an open subset of immersions.
We can detail and prove this fact in two ways.(i)We can see it as a corollary of Lemma 8. Consider a neighborhood defined using the tubular coordinates; precisely, define as in the second definition in Section 3.7.6, choosing and ; knowing that , we could choose , but then by Lemma 8, would be the identity. So, each and any curve in is freely immersed. As discussed in Section 3.7.6, this proves the result in the manifold of smooth immersed curves, where the above neighborhoods define a topology.(ii)If we instead want to prove this for the standard Banach topology (first definition in Section 3.7.6), we can proceed as follows. Suppose that is a sequence of immersed curves that are not free; and suppose that in . We may rescale and reparametrize all the curves so that all have length and , and still in ; we skip the details (see Theorem 7 in [20]); these assumptions simplify the following arguments. Let be a sequence such that and is not the identity; as above, we know that and we may choose to be a generator of the isotropy group, so that ; we know that is bounded by 4 times the curvature, so up to a subsequence is constant, let us call it , and then with , and passing to the limits , so is not freely immersed.
Note that both proofs need a compactness argument; this seems unavoidable since the size of the neighborhood cannot be estimated by using differentiable quantities, as explained in Example 9.
4. The Manifold of Free Geometric Curves
Definition 18 (classes of curves). (i) is the class of immersed curves : curves such that at all points(ii) is the class of freely immersed curve, the immersed such that, moreover, if is a diffeomorphism and for all , then (iii) are the embedded curves, maps that are diffeomorphic onto their image ; and the image is an embedded submanifold of of dimension 1
Each class contains the one following it (this follows from the propositions seen in Section 2.3).
4.1. Proof of Theorem 1
Definition 19. is the quotient of (free immersions) by the positive diffeomorphisms (reparameterization).
We now provide the complete proof of Theorem 1, namely, that this is a manifold, for the case of smooth freely immersed planar curves; afterward, we will show in Section 4.2 how and where the proof in [5] misses some key arguments.
The following proof is for immersed curves , in a forthcoming paper [15], we will explain how it can be generalized to the case of immersions .
4.1.1. Quotient Topology
We discuss the topological aspect of Theorem 1.
Let be the canonical projection of the positive quotient that defines in (157).
The definition of the quotient topology is as follows. A set is open in when the union of its orbitsis open in , that is, it is open in .
We endow with the topology described earlier and with the induced quotient topology.
Now, we want to describe a specific family of open neighborhoods that will be quite useful. Fix smooth freely immersed. Let , where was defined in Lemma 8.
Proposition 11. Consider the setand this set is open in .
Proof. This is a simple case of the arguments of Section 3.7.6. Let , let andwe know that ; if is a smooth curve and satisfiesthen by the results in Section 3.
By Theorem 4, all curves in are freely immersed.
Now, let us reparameterize all the curves in and definesince the above conditions are reparameterization invariant, thenthat is an union of open sets; hence, it is open in . Moreover, it contains all the orbits of all of its curves: in the language of [5], we may say that “ is saturated for the action of Diff .”
So, we defineand we have
Hence, is open in .
4.1.2. Geometric Representation
We discuss the representation aspect of Theorem 1. Consider again the set defined in (162). For any curve in this set, by Theorem 3, we have a representationwith .
Let for convenience; by the derepresentation result Theorem 10 (setting in that proposition and noting ), we haveso there is an unique reparameterization of that can be expressed in tubular coordinates around ; by Lemma 8, this means that is uniquely identified by ; so, we will concentrate on .
Proposition 12. Letbe the set of all such . This set is open.
Proof. The map is smooth, and is the projection on the second component of the counterimage of that is open.
All of the above can be stated in the language of [5] as follows: the set is an open neighborhood of in , it is composed only of freely immersed curves, it is saturated for the Diff -action and the mapsplits it smoothly as
4.1.3. Charts
Choose a curve ; consider the map (it is not the same map defined in Proposition 7)and we already proved in Lemma 8 that it is injective; is also smooth as a map from to .
If we composethen the compositionis bijective: indeed, if , then we proved in Section 3 that, picking a in the equivalence class ,for an unique , and thenfor a unique .
4.1.4. Atlas
To conclude, we discuss the Atlas of charts needed for Theorem 1. For , consider now two equivalence classes and choose a curve in each; we consider the maps
We want to check that these are charts of an Atlas for the manifold.
Suppose thatthenand we need to check thatis smooth in a neighborhood of . We can change variable in the previous one, that is, reparameterize , so thatbut for near we know thatby Lemma 10, so by the representation Theorem 3, there are and dependent on such that
The representation theorem’s proof shows that the dependency of on is smooth (it is given by the inverse of the tubular coordinates, as discussed in the nearby projection Lemma 5).
This concludes the desired proof of Theorem 1.
4.2. Comparison with [5]
We endow with a Riemannian metric such that scalar curvatures are bounded and convexity radius is bounded uniformly away from zero; see [7].
The proof in [5] is presented for generic immersions ; we fix one such immersion.
We follow notations and definitions in [5], here copied for convenience of the user (parts copied from [5] will be in italic, and enclosed in ≪...≫).
≪We choose connected open sets and such that , is an open cover of , each is compact, is a locally finite open cover of , and such that is an embedding.≫
≪Let be the normal bundle of , defined in the following way: for , letthat is, the orthogonal complement of tangent at of the immersed manifold in .≫
≪The following diagram
is a vector bundle homomorphism over , which is fiberwise injective.≫
≪Let be the exponential map on . Now, there is a neighbourhood of the zero section in the previous bundle that is small enough so thatwhen restricted to is a diffeomorphism with its image. The restriction of to is called . It will serve us as a substitute for a tubular neighborhood of .≫
The notation is not described in [5], but by its usage it should be equivalent to . Note also that, later on, the paper adds the superscript to and will write it as .
Consider planar immersed curves : normal vectors at are a one dimensional space (where is the normal vector to the curve as defined in Definition 12); hence, the fibre of is one dimensional, so a point in the bundle can be represented by a pair , and the map becomes the map defined in (52) in Proposition 7. Fix a , small, as we will discuss later on. We cover by arcs each shorter than , and subarcs that can be chosen so that they are an open cover; for small we can ensure that tubular coordinates can be perused. The open set will include normal vectors with .
The statement of main Theorem 1.5 in [5] starts as follows.
Let be a free immersion . Then, there is an open neighborhood in Imm which is saturated for the Diff -action and which splits smoothly as
Here, is a smooth splitting submanifold of Imm , diffeomorphic to an open neighborhood of 0 in . In particular, the space is open in .
The proof covers also the case when is not compact; we will assume that is compact so that some arguments can be simplified.
The proof goes as follows.
Define(the proof then goes on showing that this is an open set—we skip details).
For each , we define
Indeed, we know that is a diffeomorphism onto its image, and that when , by definition of .
Then is a mapping which is bijective onto the open setin . Its inverse is given by the smooth mapping
The proof then goes on showing that this is smooth (we skip details).
We now translate the above objects into the language of Section 4.1. If is a freely immersed curve, and is a curve such thatthen ; so, for small, for each , there are such thatand this pair is exactly associated to correct point in the above bundle, that is, we can write
Possibly reducing the width of the tubular neighborhoods, we can also use Proposition 9 to ensure that the above representation is “unique.” So, Proposition 9 can be applied, and this means that, for each , we can writeso, in conclusion, we can explicitly write the map above defined aswhere the first component in encodes a position in the base space of the bundle , the second encodes a normal vector to the curve at . Indeed, the inverse is exactly the map (193).
The proof continues as follows.
We have for those which are near enough to the identity so that (that was defined in (188)). We consider now the open set (we added the notation for ease of reference)
Obviously, we have a smooth mapping from it intogiven bywhere is the space of sections of .≫
Here comes the first mistake in the original proof. Consider Example 8, where is the blue curve and is the red curve. There is a choice of such that . We can expressand then define by (194); but when we apply the rule (197) to this , we obtain thatand is a map that is not a diffeomorphism.
The correct statement is that map (197) achieves a splitting of the open set described in (195) into
Some conditions must be added to the definition of to make sure that is a diffeomorphism; as was done in (159) to define , by adding the condition : this condition is necessary to apply the representation Theorem 4. No similar condition is present in the proof in [5].
The proof afterwards proceeds as follows.
So, if we let
We havesince the action of Diff on is free. Consequently, Diff acts freely on each immersion in , so is open in .
(this is not the same as the defined in (168), but it has the same scope; the latter one is an open neighborhood of 0 in )
This is the second mistake in the proof.
Even if we restrict the open set described in (195) by adding a first order condition, so that the map (197) properly splits , we have not guarantee that all curves in the associated neighborhood are freely immersed. This is shown in Example 9.
This is why, in our proof, we added the condition to the definition of .
We have shown that the proof in [5] does not prove the desired result.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
The author thanks Prof. Kathryn Mann for her help on the properties of .