Space of subspheres and conformal invariants of curves

# Space of subspheres and conformal invariants of curves

## Abstract

A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in the Minkowski space, which can naturally produce the conformal invariants and the normal form of the curve. We also give characterization of canal surfaces in terms of curves in the set of circles.

Key words and phrases. Conformal arc-length, conformal curvature, conformal torsion, osculating circles, osculating spheres, moving frames.

2000 Mathematics Subject Classification. 53A30, 53B30.

## 1 Introduction

A space curve is determined by the arc-length, curvature, and torsion up to motions of . In a Möbius geometric framework, it was shown by Fialkow ([F]) that a space curve is determined by three conformal invariants, conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations.

The conformal arc-length was found by Liebmann [L] for space curves and by Pick for plane curves. The conformal curvature and torsion were given by Vessiot [V]. They have been studied by Fialkow using conformal derivations [F], by Sulanke using Cartan’s group theoretical method of moving frames [Su], and by Cairns, Sharpe, and Webb using normal forms [CSW]. Conformal torsion was also studied in [MRS] using conformal invariants for pairs of spheres.

In [LO2] the first and the second authors showed that the conformal arc-length can be considered as “-dimensional measure” of the curve of osculating circles. This paper is a natural continuation of it. We use both curves of osculating circles and of osculating spheres, which belong to different spaces, and by doing so we give (hopefully) new formula for the conformal curvature (theorem 4.6). We also give moving frames in the Minkowski space using and . Then the conformal curvature and conformal torsion appear in Frenet matrix (this is a just of translation of Sulanke’s result [Su] to our context). By taking the projection to the Euclidean space which can be realized in as an affine section of the light cone, we obtain the normal form given in [CSW]. We give a proof of formulae of conformal curvature and torsion in [CSW] using our theorem 4.6.

Thus, we get geometric and simpler description of integration of preceding studies of conformal invariants of space (and planar) curves.

We also study canal surfaces and give the characterization of them as curves in the set of circles in .

The authors thank Gil Solanes for many useful discussions.

## 2 Preliminars

Let us start with basics in Möbius geometry which are needed for the study of curves of osculating circles and of osculating spheres.

### 2.1 Realization of R3 and \boldmathS3 in Minkowski space R51

We start by recalling a commonly used models of a sphere and the Euclidean space in Möbius geometry (cf. [B, Ce, HJ]). Let us explain in a -dimensional case.

The Minkowski space is endowed with an indefinite inner product (the Minkowski product or the Lorentz form) given by

 ⟨x,y⟩=−x0y0+x1y1+x2y2+x3y3+x4y4.

The light cone is given by . A vector subspace of is said to be time-like if it contains a non-zero time-like vector. When , is time-like if and only if it intersects the light cone transversally.

A -sphere or can be identified with the projectivization of the light cone. In fact, they can be isometrically embedded in as the intersection of the light cone and a codimension affine subspace . When can be expressed as for some unit time-like vector , in which case is tangent to the hyperboloid at point , is a unit sphere. When can be expressed as for some light-like vector , in which case is parallel to a codimension subspace which is tangent to the light cone in , is a paraboloid . The metric induced from the Lorentz quadratic form on is Euclidean. For example, when , the Euclidean space can be isometrically embedded as

 R3∋\boldmathx↦(1√2+\boldmathx⋅\boldmathx2√2,−1√2+\boldmathx⋅\boldmathx2√2,% \boldmathx)∈E3=C∩{x|⟨x,n⟩=−1}.

We use the notation and to emphasize that they are embedded in .

### 2.2 de Sitter space as the set of codimension 1 spheres

An oriented sphere in can be obtained as the intersection of an oriented time-like -dimensional subspace of . Therefore the set of oriented spheres in can be identified with the Grassmann manifold of oriented time-like dimensional subspaces of . By taking the orthogonal complement, we obtain a bijection between and the set of oriented space-like lines, which can be identified with the quadric called de Sitter space (Figure 1). The bijection from to is given by

 Λ4∋σ↦Σ=S3∩(span(σ))⊥∈S(2,3),

oriented as the boundary of the ball , and its inverse is given by

 S(2,3)∋S3∩span(u1,u2,u3,u4)↦u1×u2×u3×u4∥u1×u2×u3×u4∥∈Λ4,

where is the Lorentz vector product with respect to that is characterized by , the norm being equal to the volume of parallelepiped spanned by which is given by , and ([LO]3). Then is given by and . Direct computation shows

 ⟨u1×u2×u3×u4,v1×v2×v3×v4⟩=−det(⟨ui,vj⟩) (2.1)

Similarly, the set of oriented circles in can be identified with de Sitter space in .

### 2.3 Pseudo-Riemannian structure of indefinite Grassmann manifolds

In general, the set of oriented -dimensional subspheres in can be identified with the Grassmann manifold of oriented time-like -dimensional subspaces in . By taking the orthogonal complement we obtain a bijection from to the Grassmann manifold of oriented space-like -dimensional subspaces in . Thus we need the pseudo-Riemannian structure of these Grassmannians.

The canonical extension of positive definite inner product to Grassmann algebras is given by

 ⟨u1∧⋯∧uq,v1∧⋯∧vq⟩=det(⟨ui,vj⟩).

But in our case, as we start with the Minkowski space, it does not fit (2.1). Therefore we agree (after [HJ] page 280) that the pseudo-Riemannian structures on our Grassmannians are given by

 ⟨u1∧⋯∧uq,v1∧⋯∧vq⟩={−det(⟨ui,vj⟩)if span(ui) and span(vi) are time-% like,−det(⟨ui,vj⟩)%if$span(ui)$and$span(vi)$arespace−like. (2.2)

### 2.4 Two Grassmannians as the set of oriented circles in S3

An oriented circle in can be obtained as the intersection of an oriented time-like -dimensional subspace of . Therefore the set of oriented circles in can be identified with the Grassmann manifold of oriented time-like dimensional subspaces of . By taking the orthogonal complement, we obtain the bijection between and the Grassmann manifold of oriented space-like dimensional subspaces of .

Recall that the wedge product of two vectors in , and , is given by , where the Plücker coordinates are given by

 pij=∣∣∣uiujvivj∣∣∣, (2.3)

and a vectore in is a pure -vector, i.e. the wedge product of two vectors in if and only if satisfy the Plücker relation:

 p01p23−p02p13+p03p12 = 0, (2.4) p01p24−p02p14+p04p12 = 0, (2.5) p01p34−p03p14+p04p13 = 0, (2.6) p02p34−p03p24+p04p23 = 0, (2.7) p12p34−p13p24+p14p23 = 0. (2.8)

We remark that all of them are not independent. For example, the relations (2.7) and (2.8) can be derived from the rest if .

Then can be identified with the set of unit space-like pure -vectors:

 ˜G+2,5≅{\boldmathp=(pij)0≤i

Now the identification between and can be explicitly given by

 S(1,3)∋Γ=S3∩(span(u,v))⊥↦u∧v∥u∧v∥∈˜G+2,5. (2.9)

### 2.5 How to express osculating circles and osculating spheres

Let be a point in a curve in , where is the arc-length parameter. We always assume that the differential of never vanishes in what follows. Let be a time-like vector subspace of of dimension (or ). Then the curve has contact of order with the circle (or the sphere respectively) if and only if belong to (remark that the curve is not in but in ). Therefore, an osculating circle to at is given by , and an osculating sphere is generically given by .

In what follows we assume that the curve is vertex free, i.e. the osculating circles have contact of order exactly equal to . Then the point in that corresponds to the osculating sphere at point is given by

 σ(s)=m(s)×m′(s)×m′′(s)×m′′′(s)∥m(s)×m′(s)×m′′(s)×m′′′(s)∥. (2.10)

Note that can be expressed as

 σ′(s)=m(s)×m′(s)×m′′(s)×(am′′′(s)+bm(4)(s)) (2.11)

for some . Let us further assume that never vanishes. As we have . The formulae (2.10) and (2.11) imply

 span(σ(s),σ′(s))=(span(m(s),m′(s),m′′(s)))⊥.

Therefore, the osculating circle to at point corresponds to a point in given by

 γ(s)=±σ(s)∧σ′(s)∥σ(s)∧σ′(s)∥.

For simplicity’s sake, we assume that the sign is in what follows. If we denote the derivative by the arc-length parameter of the curve of osculating spheres in by putting above, we have

 γ=σ∧\Huge.σ (2.12)

An osculating circle to a curve in (in this case, the osculating sphere is constantly equal to ) can be expressed in a similar way as (2.10). In this case, as ([LO2]), a point in which corresponds to the osculating circle is given by .

Similarly, an osculating circle to a curve in can be expressed by

 γ(s)=m(s)∧m′(s)∧m′′(s)∈˜G−3,5. (2.13)

### 2.6 Our notations and assumptions

We express a curve in or by and a point on it by . We always assume that is vertex-free. The letters and express the osculating circle and sphere respectively unless otherwise mentioned. There are one or two natural conformally invariant parameters: one is the conformal arc-length which will be denoted by , and the other is the arc-length parameter of the curve of osculating spheres which will be denoted by . The latter appears only when is a space curve. We also use the arc-length of . In order to avoid confusion, we use different notations to express the derivatives by these parameters. The derivatives by , and are expressed by putting , and respectively. For simplicity’s sake, we put a technical assumption that never vanishes when is a space curve.

### 2.7 Null curve of osculating circles and conformal arc-length

Let us first consider a smooth one parameter family of circles in given by their centers and radii . Locally speaking, these circles may admit an envelope formed of two curves, or exceptionally one curve; this is the case we will consider here, because we consider the family of osculating circles to a curve.

Then everywhere. The fact that the osculating circles of a planar arc with monotone curvature are nested along the arc as is illustrated in Figure 2 was observed in the beginning of th century by Kneser ([K]).

In our language,

###### Lemma 2.1

([LO2]) A curve of osculating circles to a curve in is light-like, and the point on can be given by .

Similarly, when is a curve in , a curve of osculating circles is a null curve in with being a pure vector. If is a plane in which corresponds to , then it is tangent to the light cone in a line .

###### Proposition 2.2

([LO2]) Let denote a curve of osculating circles to a curve in or . Then the -form is independent of the parameter . Let be a parameter so that the . In other words, the parameter can be characterized by

 ⟨\tiny∘∘γ,\tiny∘∘γ⟩=1, (2.14)

where . It can be uniquely determined up to for some constant .

This parameter is called the conformal arc-length of . Our assumption that the curve is vertex-free guarantees that serves as a non-singular parameter of .

## 3 Curves in E2 or S2

Let be an osculating circle to a curve in at a point . Then they form a curve in de Sitter space . We express the derivative by the conformal arc-length by putting above.

### 3.1 The moving frame and Frenet formula

Our moving frames consist of two space-like vectors and two light-like vectors, instead of three space-like vectors and a time-like vector, since we take a light-like vector in as our first frame, which we denote by . Another light-like vector, which comes last in our frames, is chosen so that . The middle two space-like vectors of our frames are taken from an orthonormal basis of . The moving frames of this form are called isotropic orthonormal frames.

Let us choose the first vector of our moving frames in so that . Then we can take . Our second vector is . Then the -coordinate of the normal form can be given by . We choose our third vector in so that and that the coordinate of the normal form can be given by . The -axis of the normal form should be the osculating circle. Therefore, we can take . Our last vector is a light-like vector in that satisfies .

Put (cf. [Su])

 Q2=−12⟨\tiny∘∘n,\tiny∘∘n⟩. (3.1)

Then . Lemma 2.1 and proposition 2.2 imply that the for small are given by table 1. It implies that our last frame is given by .

We remark that our moving frames

 n=\tiny∘γ, v1=\tiny∘n=\tiny∘∘γ, v2=γ,  n∗=−Q2n+\tiny∘∘n=−Q2\tiny∘γ+\tiny∘∘∘γ

also serve as moving frames of the curve of osculating circles (Remark that is not necessarily equal to a point in the curve ).

The Frenet formula with respect to our moving frames is given by

 ddρ⎛⎜ ⎜ ⎜⎝nv1v2n∗⎞⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜⎝0100Q200110000Q210⎞⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜⎝nv1v2n∗⎞⎟ ⎟ ⎟⎠. (3.2)

See remark 4.8.

### 3.2 Normal form

The normal form of a planar curve can be obtained from that of a space curve (see subsection 4.3) by putting and and forgetting the -coordinate. If we only look for the normal form of a plane curve, the computation is simpler. Thanks to table 1, we do not have to rewrite in terms of the frames. The direction vectors of - and - axes at are given by and respectively.

## 4 Curves in E3 or S3

### 4.1 Osculating spheres and conformal torsion

Let be a point on a curve in . Let be a curve of osculating circles, and a curve of osculating spheres. Since an osculating sphere intersects an infinitesimally close osculating sphere in an osculating circle, is a space-like curve in . Let be the conformal arc-length and the arc-length parameter of the curve . We express the derivatives by and by putting above and respectively. Put

 T=dldρ. (4.1)

As measures the infinitesimal angle variation, measures how an osculating sphere rotates around an osculating circle with respect to the conformal arc-length. For simplicity’s sake, let us assume in what follows. It is proved in [RS] that coincides with the conformal torsion up to sign. As is pointed out in [Sh], the conformal torsion can be determined up to sign. We remark that the conformal torsion is identically equal to if and only if is a planar or a spherical curve.

###### Lemma 4.1

We have .

Proof: Since and , we have and . On the other hand, since we have . Therefore, the formula (2.2) implies

 ⟨\Huge.γ,\Huge.γ⟩=∣∣ ∣ ∣∣⟨σ,σ⟩⟨σ,\Huge.\kern-3.0pt.σ⟩⟨σ,\Huge.\kern-3.0pt.σ⟩⟨\Huge.\kern-3.0pt.σ,\Huge.\kern-3.0pt% .σ⟩∣∣ ∣ ∣∣.

As by lemma 2.1, it implies .

###### Proposition 4.2

The conformal torsion satisfies

 T=14√⟨\Huge.\kern-3.0pt.γ,\Huge.\kern-3.0pt.γ⟩=14√⟨\Huge.\kern-3.0pt.\kern-3.0pt.σ,\Huge.\kern-3.0pt.\kern-3.0pt.σ⟩−1.

Proof: The first equality comes from proposition 2.2, the fact that for any parameter ([LO2]).

Since , we have and hence and . Therefore the formula (2.2) implies

 ⟨\Huge.σ∧\Huge.\kern-3.0% pt.σ,\Huge.σ∧\Huge.% \kern-3.0pt.σ⟩=1,⟨\Huge.σ∧\Huge.\kern-3.0pt.σ,σ∧\Huge.\kern-3.0pt.\kern-3.0pt.σ⟩=−1, and ⟨σ∧\Huge.\kern-3.0pt.\kern-3.0pt.σ,σ∧\Huge.\kern-3.0pt.\kern-3.0pt.σ⟩=⟨\Huge.\kern-3.0pt.\kern-3.0pt.σ,\Huge.\kern-3.0pt.\kern-3.0pt.σ⟩.

As and therefore , we have , which completes the proof.

It follows that for small are given by table 2.

###### Lemma 4.3

Under our assumption, i.e. if the curve is vertex-free and never vanishes, the five vectors , and are linearly independent.

Proof: Suppose for some . Put and . Then, by taking pseudo inner product with , we have

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝10−101010−10−1010−G30−10G312\Huge.G310−G312\Huge.G3G4⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝a0a1a2a3a4⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=\boldmath0.

The determinant of the coefficient matrix is equal to .

### 4.2 Moving frame in R51 and Frenet formula

###### Proposition 4.4

([Y]) A point on the curve can be expressed in terms of the osculating spheres as

 m=c(σ+\Huge.\kern-3.0pt.σ)(c∈R∖{0}). (4.2)

Proof: Since and by lemma 4.1 and its proof, is a light-like vector.

On the other hand, as can be expressed as for some function , can be expressed as a linear combination of vectors of the form . Therefore, , which completes the proof, as is tangent to the light cone in .

We choose the conformal arc-length, not , for the parameter of the Frenet formula, as is the case in most preceding studies. Let us choose the first vector of our moving frames in so that . Then we can take . Our second vector is . Then the -coordinate of the normal form can be given by . We choose our third and fourth vectors , and in so that and that the , and coordinates of the normal form can be given by , and respectively. The -plane of the normal form should be the osculating sphere. Therefore we can take . The choice of the frame is important to get the same normal form as in [CSW]. We put here so that the Frenet matrix and the normal form fit with those in [Su] and [CSW] respectively. A sphere which corresponds to should intersect the osculating sphere orthogonally in the osculating circle. Therefore we can take . Our last vector is a light-like vector in that satisfies .

We remark that our moving frames

 n=T(σ+\Huge.\kern-3.0pt.σ),v1=\tiny∘n=T\Huge.T(σ+% \Huge.\kern-3.0pt.σ)+T2(\Huge.σ+\Huge.\kern-3.0pt.\kern-3.0pt.σ),v2=\Huge.σ,v3=−σ,n∗

also serve as moving frames of the curve of osculating spheres. Note that is not necessarily equal to a point in the curve . In order to have , we enlarge or shrink the point vector of a curve in the light cone keeping the parameter to be the conformal arc-length.

###### Proposition 4.5

([Su]) Put

 Q=⟨\tiny∘n,\tiny∘n∗⟩=−⟨\tiny∘∘n,n∗⟩. (4.3)

Then the Frenet formula with respect to our moving frames is given by

 ddρ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝nv1v2v3n∗⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝01000Q0001100T000−T000Q100⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝nv1v2v3n∗⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (4.4)

Proof: The formula (4.1) implies for any . The first, third, and fourth rows of the matrix follow from the definition of the frames. The other two rows can be obtained by derivating the scalar products of the frames, i.e. etc. Remark that “- (or -)coordinate” of a vector can be given by (or respectively).

Since we have . Therefore,

 Q=−12⟨\tiny∘∘n,\tiny∘∘n⟩. (4.5)

This quantity is called the conformal curvature (denoted by in [Su], whereas the conformal torsion is denoted by in [Su]).

As , our last frame is given by as in the case of planar curve. Thus our isotropic orthonormal moving frames are

 n∈span(m),v1=\tiny∘n,v2=\tiny∘σ∥\tiny∘σ∥,v3=−σ,n∗=−Qn+\tiny∘∘n.
###### Theorem 4.6

Let be a curve of osculating circles and a curve of osculating spheres. Then the conformal curvature and the conformal torsion satisfy

 T = √⟨\tiny∘σ,\tiny∘σ⟩, (4.6) Q = −12⟨\tiny∘∘∘γ,\tiny∘∘∘γ⟩+3⟨\tiny∘σ,\tiny∘σ⟩. (4.7)

Proof: The first equation is trivial from the definition (4.1) of as .

Since , we have

 \tiny∘γ=n∧v3\tiny∘∘∘γ=−2Tv1∧v2−\tiny∘Tn∧v2+(Q−T2)n∧v3−v3∧n∗.

It follows that