Neighborhoods and Manifolds of Immersed Curves
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.
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  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  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 ; for example  and a survey in .
An unjustified feature of the model of  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 , using a result from .
A purpose of this paper is to revisit the key result in : 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  was incorrect.
A supplemental file contains Wolfram Mathematica code to generate some of the figures.
In this section, we will present well-known definitions and results regarding immersions, with particular attention to immersed curves.
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 , that coincides with the compact-open -topology as defined in 41.9 in ; 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  coincides with the Whitney -topology as defined in 41.10 in 
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 ) that is the strict inductive limit (for the definition of strict inductive limit and its properties, we refer to 17G at page 148 in ) 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 .
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 .
We will omit the superscript “” from in the following, for ease of notation.
2.2.1. Free Immersion
Definition 3. An immersion is called “free” if for implies that is the identity.
Proposition 3 (see  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  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  (see  for a self-contained presentation); it is also a simple group: see Discussion in Sec. 2 in  for further references. (The author thanks Prof. Kathryn Mann for her help on these subjects.)
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 ). 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 .
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 .
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  or Theorem 53.1 in , 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
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 . (We remark that the theorem in  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  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 .
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,
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 ). 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).
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 ).
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  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  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.
3.7.1. 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