J. Math. Anal. Appl. 473 (2019) 112–140
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Journal of Mathematical Analysis and Applications
www.elsevier.com/locate/jmaa
Conformal theory of curves with tractors
Josef Šilhan a,∗
, Vojtěch Žádník b
a
Institute of Mathematics and Statistics, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
b
Faculty of Education, Masaryk University, Poříčí 31, 60300 Brno, Czech Republic
a r t i c l e i n f o a b s t r a c t
Article history:
Received 23 July 2018
Available online 22 December 2018
Submitted by J. Lenells
Keywords:
Conformal geometry
Tractor calculus
Curves
Invariants
We present the general theory of curves in conformal geometry using tractor
calculus. This primarily involves a specification of distinguished parametrizations
and relative and absolute conformal invariants of generic curves. The absolute
conformal invariants are defined via a tractor analogue of the classical Frenet frame
construction and then expressed in terms of relative ones. Our approach applies
likewise to conformal structures of any signature; in the case of indefinite signature
we focus especially on the null curves. It also provides a conceptual tool for handling
distinguished families of curves (conformal circles and conformal null helices) and
conserved quantities along them.
© 2018 Elsevier Inc. All rights reserved.
1. Introduction
The local geometry of curves is a classical subject, nowadays developed for various geometric structures
using various views and methods. The study generally starts in flat spaces, the passage to general curved
cases demands new ideas and attitudes. We are going to evolve an approach based on conformal tractor
calculus. In this section, we present main sources of inspiration, summarize main results and introduce main
tools needed later.
1.1. Main sources and methods
To our knowledge, the first general treatment of curves in conformal Riemannian manifolds is due to
Fialkow [9], which is therefore used as a basic reference for comparisons. In that paper, a conformally
invariant Frenet-like approach is developed. This primarily requires a natural distinguished parametrization
of the curve (in the place of the arc-length parameter in the Riemannian setting), a natural starting object
(in the place of the unit tangent vector) and a notion of derivative along the curve (in the place of the
* Corresponding author.
E-mail addresses: silhan@math.muni.cz (J. Šilhan), zadnik@mail.muni.cz (V. Žádník).
https://doi.org/10.1016/j.jmaa.2018.12.038
0022-247X/© 2018 Elsevier Inc. All rights reserved.
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 113
restricted Levi-Civita connection). Having all these instruments, it is easy to derive a conformal analogue
of Fernet formulas with a distinguished set of conformal invariants, the conformal curvatures of the curve.
For generic curves in an n-dimensional manifold, this yields n − 1 conformal curvatures, one of which has
an exceptional flavour.
The Frenet frame can be seen as an example of a more general concept of moving frame by Cartan.
An application of the latter method for immersed submanifolds, especially curves, in the homogeneous
Möbius space is presented by Schiemangk and Sulanke in [16] and [20], which we adduce as important
sources of inspiration. That way, the curve in the homogeneous space is covered by a curve in the principal
group so that the restriction of the Maurer–Cartan form to the lift reveals the generating set of invariants.
The lift can be identified with, and practically is constructed as, a curve of (pseudo-)orthonormal frames in
the ambient Minkowski space, whose dimension is two more higher than the dimension of the Möbius space.
Both the homogeneous principal bundle with the Maurer–Cartan form and the associated homogeneous
vector bundle, whose fibre is the ambient Minkowski space with the induced linear connection, has a counterpart
over general conformal manifolds: the former leads to the notion of conformal Cartan connection,
the latter to the Thomas standard tractor bundle with its linear connection and parallel bundle metric.
These two approaches are basically equivalent, in this paper we exploit the latter one. One of its main
advantages is that everything can be set off pretty directly in terms of underlying data, while it still has
very conceptual flavour. This, on the one hand, allows one to easily adapt key ideas from the homogeneous
setting to the tractorial one. On the other hand, expanding any tractorial formula yields a very concrete
tensorial expression that allows comparisons to results obtained by that means. We refer to the essential
work [2] by Bailey, Eastwood and Gover both for the generalities on tractor calculus and for an initial step
in the study of (distinguished) curves in this manner.
Besides the just cited main sources, there is a wide literature discussing curves in conformal and other
geometries from various aspects. A short review of the development of the topic for conformal structures
can be found in the introduction of [5]. Another references that are close to our purposes are [3], [4] and [13].
An exhibition of tensorial techniques in the study of curves in related geometries can be found in [14]. For
typical subtleties in dealing with null curves in pseudo-Riemannian geometry, see e.g. [7]. For an application
of the Cartan’s method of moving frame to curves in a broad class of homogeneous spaces, see [6].
1.2. Aim, structure and results
We study the local differential geometry of curves in general conformal manifolds of general signature
using the tractor calculus. Although the subject is indeed classic, the systematic tractorial approach is novel.
This approach has not only obvious formal benefits, but also a potential for discovering new results and
relations. In particular, the discussion for null curves in the case of indefinite signature (for which almost
nothing is known) is very parallel to the one in positive definite case and leads to results that we, at least,
would never obtain by other means.
The rough structure of the paper is as follows: In sections 2 and 4 we develop the theory for conformal
Riemannian structures, in section 5 we discuss its analogies in indefinite signature. We primarily deal with
generic curves, whereas special cases are briefly mentioned in accompanying remarks. However, the most
special case—conformal circles—is dealt individually in section 3.
In section 2 we follow the setting of [2], where one already finds the tractorial determination of the preferred
class of projective parametrizations on any curve as well as the projectively parametrized conformal
circles. Continuing further with tractors of higher order, we build a natural reservoir of relative conformal
invariants associated to any curve (subsection 2.3). The simplest of these leads to the notion of conformal
arc-length, the distinguished conformally invariant parametrization of the curve (subsection 2.4). This invariant
vanishes identically along the curve if and only if the curve is an arbitrarily parametrized conformal
circle (Proposition 3.3). As a demonstration of the ease of use of tractors we describe some conserved quan-
114 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
tities along projectively parametrized conformal circles on manifolds admitting an almost Einstein scale,
respectively normal conformal Killing field (subsection 3.2).
In section 4 we launch the Frenet-like procedure to absolute conformal invariants of curve:
• construct a pseudo-orthonormal tractor Frenet frame along the given curve,
• differentiate with respect to the conformal arc-length and extract the tractor Frenet formulas,
• the coefficients of that system determine the generating set of invariants.
For the curve in an n-dimensional manifold, we have n − 1 conformal curvatures, one of which has an
exceptional flavour. The vanishing of the exceptional curvature has an immediate interpretation relating the
above mentioned parametrizations (Proposition 4.3). By construction, all conformal curvatures are expressed
via the tractors from the tractor Frenet frame, i.e. with respect to the conformal arc-length parametrization.
Alternative expressions in terms of initial tractors are also possible. In particular, for all nonexceptional
curvatures, we have very simple formulas using the previously defined relative conformal invariants, i.e. with
respect to an arbitrary parametrization (Theorem 4.6). For a given scale, we also indicate how to express
conformal curvatures in terms of the Riemannian ones (subsection 4.3). Besides the pure pleasure, this effort
allows a comparison of our invariants with those in the literature, especially in [9].
In section 5 we adapt the previous scheme to conformal manifolds of indefinite signature. In that case
we distinguish space-, time- and light-like curves according to the type of their tangent vectors. A full
type classification of curves (via associated tractors) becomes very rich, depending on the dimension and
signature. The construction of the tractor Frenet frame has to be adapted to the respective type, which makes
any attempt on its universal description impossible. Avoiding the complicated branching of the discussion,
we only point out the main features (Remark 4.4(3)) and focus on the light-like curves. Among these
curves we identify an appropriate analogue of conformal circles, the conformal null helices. Notably, in their
characterization another family of relative conformal invariants of Wilczynski type appears (Theorem 5.8).
More details on the construction of the tractor Frenet frame and expressions of the corresponding conformal
curvatures are discussed in the case of Lorentzian signature (subsection 5.4).
1.3. Notation and conventions
Most of the following conventions is taken from [2]. A conformal structure of signature (p, q) on a smooth
manifold M of dimension n = p+q is a class of pseudo-Riemannian metrics of signature (p, q) that differ by
a multiple of an everywhere positive function. For all tensorial objects on M we use the standard abstract
index notation. Thus, the symbol μa
and μa refers to a section of the tangent and cotangent bundle, which
is denoted as Ea
:= TM and Ea := T∗
M, respectively, multiple indices denote tensor products, e.g. μa
b
is a section of Ea
b
:= T∗
M ⊗ TM etc. Round brackets denote symmetrization and square brackets denote
skew symmetrization of enclosed indices, e.g. sections of E[ab] = T∗
M ∧ T∗
M are 2-forms on M. By E[w]
we denote the density bundle of conformal weight w, which is just the bundle of ordinary (−w
n )-densities.
Tensor products with another bundles are denoted as Ea
[w] := Ea
⊗ E[w] etc. In what follows, the notation
as μa
∈ Ea
[w] always means that μa
is a section (and not an element) of Ea
[w], global or local according to
the context.
Conformal structure on M can be described by the conformal metric gab which is a global section of
E(ab)[2]. Any raising and lowering of indices is provided by the conformal metric, e.g. for μa
∈ Ea
[w] we have
μa = gabμb
∈ Ea[w + 2]. A conformal scale is an everywhere positive section of E[1]. The choice of scale
σ ∈ E[1] corresponds to the choice of metric gab ∈ E(ab) from the conformal class so that gab = σ−2
gab. The
corresponding Levi-Civita connection is denoted as ∇. The Schouten tensor, which is a trace modification of
the Ricci tensor, is denoted as Pab. Transformations of quantities under the change of scale will be denoted
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 115
by hats. In particular, for σ = fσ, Υa = f−1
∇af and any μa
∈ Ea
, the Levi-Civita connection and the
Schouten tensor change as
∇aμb
= ∇aμb
+ Υaμb
− μaΥb
+ μc
Υcδa
b
, (1)
Pab = Pab − ∇aΥb + ΥaΥb − 1
2 ΥcΥc
gab. (2)
Let Γ ⊂ M be a curve with a parametrization t : Γ → R and the tangent vector Ua
so that Ua
∇at = 1.
Regular curve is called space-like, time-like, respectively light-like (or null), in accord with the type of its
tangent vector Ua
, i.e. if UcUc
> 0, UcUc
< 0, respectively UcUc
= 0. Of course, the curve may change the
type of its tangent vectors. Since the whole study that follows is of very local nature, we restrict ourselves
only to the (segments of) curves of fixed type. In the first two cases, the density
u := |UcUc| ∈ E[1] (3)
is nowhere vanishing and will be employed later. Henceforth we always assume Γ is smooth. Along the curve
we use notation d
dt := Uc
∇c and
U a
:= d
dt Ua
, U a
:= d2
dt2 Ua
, . . . , U(i)a
:= di
dti Ua
(4)
for the derived vectors. By abuse of notation we often write Ua
∈ Ea
which should be read as Ua
∈ Ea
|Γ, and
similarly for any other quantities defined only along the curve Γ. Note that the vector U(i)a
has order i + 1
(with respect to Γ). Obviously, none of vectors in (4) is conformally invariant. For instance, the acceleration
vector transforms according to (1) as
U a
= U a
− Uc
Uc Υa
+ 2Uc
Υc Ua
. (5)
From this it follows that, for space- and time-like curves (but not for null curves), a metric in the conformal
class may be chosen so that U a
= 0, i.e. the curve is an affinely parametrized geodesic of the corresponding
Levi-Civita connection. This indicates the problem with a conformally invariant notion of osculating
subspaces. Instead, we are going to make use of tractors.
The conformal standard tractor bundle T over the conformal manifold M of signature (p, q), where
p + q = n = dim M, is the tractor bundle corresponding to the standard representation Rp+1,q+1
of the
conformal principal group O(p + 1, q + 1). Specifically, T has rank n + 2 and for any choice of scale, it is
identified with the direct sum
T = E[1] ⊕ Ea
[−1] ⊕ E[−1]
whose components change under the conformal rescaling as
σ
μa
ρ
=
⎛
⎝
σ
μa
+ Υa
σ
ρ − Υcμc
− 1
2 ΥcΥc
σ
⎞
⎠ . (6)
Here Υa is the 1-form corresponding to the change of scale as before. Note that the projecting (or primary)
slot is the top one. The bundle T is endowed with the standard tractor connection ∇, the linear connection
that is given by
∇a
σ
μb
ρ
=
∇aσ − μa
∇aμb
+ δa
b
ρ + Pa
b
σ
∇aρ − Pacμc
.
116 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
It follows this definition is indeed conformally invariant. The bundle T also carries the standard tractor
metric, the bundle metric of signature (p + 1, q + 1) that is schematically represented as
0 0 1
0 gab 0
1 0 0
,
i.e., for any sections U = (σ, μa
, ρ) and V = (τ, νa
, π) of T ,
U · V = μaνa
+ σπ + ρτ.
It follows the standard tractor metric is parallel with respect to the standard tractor connection.
2. Relative conformal invariants
Throughout this section we consider conformal structures of positive definite signature. The discussion
for space- and time-like curves in indefinite signature shows only minor differences, the null case is more
involved, see section 5. In the first two subsections we only slightly extend the setting of [2, section 2.8].
Then we introduce a natural family of relative conformal invariants and the notion of conformal arc-length
parameter of curve.
2.1. Canonical lift and initial relations
In the positive definite signature, any vector is space-like. Hence, for a regular smooth curve Γ ⊂ M with
a fixed parametrization t, the density (3) becomes u =
√
UcUc and it is nowhere vanishing. This provides
a lift of Γ to the standard tractor bundle T that is given by
T :=
0
0
u−1
. (7)
Along the curve we use the notation d
dt := Uc
∇c and
U := d
dt T , U := d2
dt2 T , . . . , U(i)
:= di+1
dti+1 T (8)
for the derived tractors. The notation is chosen so that the highest order term in the middle slot of U(i)
is
a multiple of U(i)a
. Note that the tractor U(i)
has order i + 2 (with respect to Γ). By construction, both
the tractor lift T and all tractors in (8) are conformally invariant objects. Explicitly, the first two derived
tractors are
U =
⎛
⎝
0
u−1
Ua
−u−3
UcU c
⎞
⎠ , (9)
U =
⎛
⎝
−u
u−1
U a
− 2u−3
UcU c
Ua
−u−3
UcU c
− u−3
UcU c
+ 3u−5
(UcU c
)2
− u−1
PcdUc
Ud
⎞
⎠ . (10)
The tractors T , U, U are linearly independent, and satisfy
T · T = 0, T · U = 0, T · U = −1,
U · U = 1, U · U = 0.
(11)
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 117
The first nontrivial identity is
U · U = 3u−2
UcU c
+ 2u−2
UcU c
− 6u−4
(UcU c
)2
+ 2PcdUc
Ud
. (12)
In order to simplify expressions, we will use the following notation
α := U · U , β := U · U , γ := U · U , etc.
Initial relations above and their consequences may be schematically indicated by the Gram matrix which,
for the sequence (T , U, U , U , U ), has the form
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 0 −1 0 α
0 1 0 −α −3
2 α
−1 0 α 1
2 α 1
2 α − β
0 −α 1
2 α β 1
2 β
α −3
2 α 1
2 α − β 1
2 β γ
⎞
⎟
⎟
⎟
⎟
⎟
⎠
. (13)
Clearly, the generating rule may be written as
U(i)
· U(i+j+1)
= U(i)
· U(i+j)
− U(i+1)
· U(i+j)
.
Later we will need some details on the next derived tractor U . As a consequence of T · U = 0, the
projecting slot of U must vanish. With the help of (12), the middle slot of U may be expressed as
u−1
U a
− 3u−3
UcU c
U a
+ 3
2 u−3
UcU c
− 3
2 u−1
(U · U ) + 2u−1
Uc
Ud
Pcd Ua
− uUc
Pc
a
.
2.2. Reparametrizations
Let ˜t = g(t) be a reparametrization of the curve Γ. All objects related to the new parameter ˜t will be
denoted by tildes in accord with d
d˜t
= g −1 d
dt , where g = dg
dt . In particular, Ua
= g −1
Ua
, u = g −1
u and
T = g T . Using just the chain rule and the Leibniz rule, one easily verifies that
U = U + g −1
g T ,
U = g −1
U + g −2
g U + (g −2
g − g −3
g 2
)T ,
U = g −2
U + 2g −2
S(g)U + g −2
S(g) T ,
(14)
where S is the Schwarzian derivative,
S(g) :=
g
g
−
3
2
g
g
2
= (ln g ) −
1
2
(ln g )
2
.
According to the starting relations (11), we have
U · U = g −2
U · U − 2S(g) ,
U · U = g −4
U · U − 4S(g)U · U + 4S(g)2
.
(15)
Hence vanishing of U · U determines a natural projective structure on any curve, cf. [2, Proposition
2.11]:
118 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Proposition 2.1 ([2]). The equation U · U = 0, regarded as a condition on the parametrization of a curve,
determines a preferred family of parametrizations with freedom given by the projective group of the line.
Any parameter from this family will be called projective.
From (15) it further follows that
U · U − (U · U )2
= g −4
U · U − (U · U )2
.
Hence the function
Φ := U · U − (U · U )2
= β − α2
(16)
is a relative conformal invariant of the curve.
2.3. Relative conformal invariants
In general, a relative conformal invariant of weight k of the curve Γ is a conformally invariant function
I : Γ → R that transforms under a reparametrization ˜t = g(t) of the curve as
I = g −k
I.
In particular, vanishing, respectively nonvanishing, of any relative invariant is independent on reparametrizations.
Conformal invariants of weight 0 are the absolute invariants.
Let us denote
Δi := det Gram T , U, U , . . . , U(i−2)
, (17)
the determinant of the Gram matrix corresponding to the first i tractors from the derived sequence
T , U, U , . . . . From (11) it is obvious that Δ1 = Δ2 = 0 and Δ3 = −1 independently of the curve and
its parametrization. These three determinants are thus absolute, but trivial conformal invariants. The first
nontrivial invariant is Δ4. From (13) it follows that
Δ4 = −β + α2
= −Φ. (18)
In general, we have
Lemma 2.2. For i = 4, . . . , n + 2, the Gram determinant Δi is a relative conformal invariant of weight
i(i − 3).
Proof. As a generalization of (14) we have
U
(j)
= g −j
U(j)
mod T , U, U , . . . , U(j−1)
,
for any j = 1, . . . , n. From this and properties of the determinant it follows that (17) changes under the
reparametrization as
Δi = g −2·2
· · · g −2(i−2)
Δi = g −i(i−3)
Δi,
for any i = 4, . . . , n + 2. 2
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 119
For i = 4, . . . , n + 2, the tractor metric restricted to T , U, . . . , U(i−2)
is nondegenerate, thus vanishing
of Δi is equivalent to the fact that the determining tractors are linearly dependent. In particular, for i ≥ 4,
vanishing of Δi implies vanishing of Δi+1. Note also that the weight of any Δi is even, thus its sign is
independent of all reparametrizations.
2.4. Conformal arc-length
By (16), respectively (18), we defined a nontrivial relative conformal invariant of the lowest possible
weight and order. It is actually nonnegative:
Lemma 2.3. Φ ≥ 0.
Proof. For a parametrization belonging to the projective family of Proposition 2.1 we have Φ = U · U .
Since the top slot of U vanishes, the previous expression equals to the norm squared of the middle slot
of U and so it is nonnegative. Since the weight of Φ is even, it is a nonnegative function independently of
the parametrization of the curve. 2
Suppose that Γ is a curve with nowhere vanishing Φ. Then
ds := 4
Φ(t) dt
is a well-defined conformally invariant 1-form along the curve whose integration yields a distinguished
parametrization of the curve; it is given uniquely up to an additive constant. Such parameter is called the
conformal arc-length. Note that if s is the conformal arc-length then Φ(s) = 1.
It will be clear from later alternative expressions that this is the same distinguished parameter as one
finds in literature, e.g. in [9] or [5]. In the later reference, one also finds a notion of vertex, which is the
point of curve where Φ = 0. The vertices of curves are clearly invariant under conformal transformations.
Generic curves are vertex-free, the opposite extreme is discussed in the next section.
Remark 2.4. For space- and time-like curves in the case of general indefinite signature, the relative conformal
invariant Φ defined by (16), respectively (18), is not necessarily nonnegative as in Lemma 2.3. It may even
happen that it vanishes although the corresponding tractors T, U, U , U are linearly independent. The
notion of vertex in such cases would rather be defined by the linear dependence of these tractors than by
vanishing of Φ, cf. Remark 3.4(2).
3. Conformal circles and conserved quantities
We continue the exposition with the assumption of positive definite signature. The only difference in
indefinite signature concerns the notion of conformal circles that are defined just for the space- or time-like
directions. Reasonable analogies for null directions are discussed in subsection 5.3.
3.1. Conformal circles
In this subsection we consider the curves for which Φ vanishes identically, i.e. the curves consisting only of
vertices. The conformal arc-length is not defined for such curves and they are excluded from the discussion
in the main part of this paper as the most degenerate cases. However, these curves form a very distinguished
family of curves that coincide with the so-called conformal circles, the curves which are in some sense the
closest conformal analogues of usual geodesics on Riemannian manifolds. In the flat case, they are just the
circles, respectively straight lines, i.e. the conformal images of Euclidean geodesics.
120 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Conformal circles are well known and studied (under various nicknames1
) in the literature. A quick survey
with an explanation of the relationship of the definition below to distinguished curves of Cartan’s conformal
connection can be found in [13, section 5.1]. Conformal circles on a general conformal manifold can also be
related to ordinary circles in the flat model via the notion of Cartan’s development of curves.
Here we adapt the notation of [1], where conformal circles are defined as the solutions to the system of
conformally invariant third order ODE’s
U a
= 3u−2
UcU c
U a
− 3
2 u−2
UcU c
Ua
+ u2
Uc
Pc
a
− 2Uc
Ud
Pcd Ua
. (19)
Contracting, respectively skew symmetrizing (19), with Ua
yields the following pair of equations equivalent
to (19):
UaU a
= 3u−2
(UcU c
)2
− 3
2 UcU c
− u2
Uc
Ud
Pcd, (20)
U[aUb] = 3u−2
UcU c
U[aUb] + u2
Uc
Pc[bUa]. (21)
From (5) we know that, for any curve with an arbitrary parametrization, a metric in the conformal class
may be chosen so that it is an affinely parametrized geodesic of the corresponding Levi-Civita connection,
i.e. U a
= 0. It is then easy to see from (20) and (21) that
Proposition 3.1 ([1]). The curve is a conformal circle if and only if there is a metric in the conformal class
such that the curve is an affinely parametrized geodesic and Uc
Pca = 0.
From the explicit expressions in subsection 2.1 we may read the following tractorial interpretations
of the equations above: The equation (19) is equivalent to vanishing of the middle slot of the tractor
U + 3
2 (U · U ) U (since the primary slot is zero, this is indeed a conformally invariant condition). This
implies that the middle slots of tractors U and U are collinear, which is just the condition (21). In
particular, this condition is independent of any reparametrization, i.e. solutions to (21) are the conformal
circles parametrized by an arbitrary parameter. The equation (20) is equivalent to U ·U = 0. In particular,
solutions to (20) correspond to the projective parameters of the curve. Considering further the injecting slot
of the tractor U , it is shown in [2, Proposition 2.12] that
Proposition 3.2 ([2]). The curve is a projectively parametrized conformal circle if and only if it obeys
U · U = 0 and U = 0. (22)
We may easily generalize this statement as follows:
Proposition 3.3. The following conditions are equivalent:
(a) the curve is a conformal circle (with and arbitrary parametrization),
(b) all its points are vertices, i.e.
Φ = U · U − (U · U )2
= 0,
(c) the rank 3 subbundle T , U, U ⊂ T is parallel along the curve.
1
E.g. they are called conformal null curves in [9] or conformal geodesics in [10] and [13]. Note that the conformal geodesics
studied in [8] or [15] form a different, more general, class of curves.
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 121
Proof. From subsection 2.1 we know that the tractors T , U, U are linearly independent for any curve and
its parametrization. From subsection 2.3 we know that Φ = 0 is equivalent to the fact that the tractors
T , U, U , U are linearly dependent, i.e. U belongs to the subbundle T , U, U ⊂ T which is therefore
parallel. Thus (b) and (c) are equivalent.
Among tractors T , U, U , U , only U has nonzero projecting slot. Hence the previous condition means
that U is a linear combination of T and U. Since the middle slot of T vanishes, the previous is equivalent
to the middle slots of U and U being collinear, which is just the condition (21). Thus (c) and (a) are
equivalent. 2
Remarks 3.4.
(1) Note the two conditions in (22) are not independent: if U = 0 then U · U = 0.
(2) Conformal circles in the case of indefinite signature exist only for space- or time-like directions. They
have all of the above stated properties except the condition (b) of Proposition 3.3, which is no more
equivalent to the others, cf. Remark 2.4.
3.2. Conserved quantities
As an application of the current approach we describe some conserved quantities (or first integrals) along
conformal circles for some specific conformal structures. Here we consider conformal structures admitting
almost Einstein scales or conformal Killing fields with some additional property. The description of the
conserved quantities is easily given due to the tractorial characterization of these additional data, which
can be found e.g. in [11].
An almost Einstein scale is a section σ ∈ E[1] satisfying the conformally invariant condition that the
trace-free part of ∇a∇bσ + Pabσ ∈ Eab[1] vanishes. Then σ is nonvanishing on an open dense set where
the metric gab = σ−2
gab is Einstein. The tractorial characterization of this fact is that the tractor field
LT
(σ) ∈ T is parallel, where LT
: E[1] → T denotes the BGG splitting operator.
Proposition 3.5. Let Γ be a conformal circle with a projective parameter t. Assume the conformal manifold M
admits an almost Einstein scale σ ∈ E[1] and let S = LT
(σ) ∈ T be the corresponding parallel tractor.
Then the function s := U · S is a conserved quantity along Γ, i.e. d
dt s = 0.
Proof. A projectively parametrized circle satisfies U = 0, the tractor S corresponding to an almost Einstein
scale satisfies S = 0. Thus d
dt (U · S) = 0. 2
A conformal Killing field ka
∈ Ea
is an infinitesimal symmetry of the conformal structure. This is
equivalent to the vanishing of the trace-free part of ∇(akb) ∈ E(ab)[2]. The tractorial characterization of this
fact is that the tractor field K := LΛ2
T
(ka
) ∈ Λ2
T satisfies
∇aK = kc
Ωca, (23)
where LΛ2
T
: Ea
→ Λ2
T denotes the BGG splitting operator and Ωcd ∈ E[cd] ⊗ Λ2
T is the curvature of the
tractor connection ∇. Further, along the given curve, we introduce the 1-form
a := Ub
Ωab(U, U ). (24)
Proposition 3.6. Let Γ be a conformal circle with a projective parameter t. Assume the conformal manifold M
admits a conformal Killing field ka
∈ Ea
satisfying ka
a = 0, and let K := LΛ2
T
(ka
) ∈ Λ2
T be the
corresponding tractor field. Then the function k := K(U, U ) is a conserved quantity along Γ, i.e. d
dt k = 0.
122 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Proof. A projectively parametrized circle satisfies U = 0. The tractor K corresponding to ka
satisfies (23)
and hence ( d
dt K)(U, U ) = ka
a = 0. Also, K is skew, thus d
dt (K(U, U )) = 0. 2
Note that the assumption ka
a = 0 in the Proposition is automatically satisfied for normal conformal
Killing fields which are characterized by the vanishing of (23).
Remarks 3.7.
(1) It is somewhat tedious to expand tractorial formulas for the conserved quantities s and k. To proceed,
one needs the respective splitting operators and some manipulation. From the explicit expressions it in
particular follows that the considered quantities are nontrivial. All that can be found already in [19].
Note also that both s and k are quantities of order 2, while the conformal circle equation is of order 3.
Further related and interesting results can be found in recent article [12].
(2) Expanding (24) yields
a = Ub
WabcdUc
U d
− 2u2
Ub
∇[aPb]cUc
,
where Wab
c
d is the conformal Weyl tensor. Remarkably, the condition a = 0 plays a role in the
variational approach to conformal circles, see [1, p. 218].
4. Absolute conformal invariants
For a generic curve, the relative conformal invariants Δi from section 2 are all nontrivial, provided that
4 ≤ i ≤ n + 2. One may therefore easily build a galaxy of absolute conformal invariants. In this section we
construct a minimal set of such invariants, which is done in a natural if not canonical way. In order to make
the construction complete, the assumption of positive definite signature is important here. In particular,
the standard tractor metric has signature (n + 1, 1).
4.1. Tractor Frenet formulas and conformal curvatures
Let Γ be a generic curve, let s be its conformal arc-length parameter and let T , U = d
ds T , U = d
ds U, . . .
be the corresponding tractors as above. The restriction of the tractor metric to the subbundle T , U, U
is nondegenerate and has signature (2, 1). Thus its orthogonal subbundle in T is complementary and has
positive definite signature. We are going to transform the initial tractor frame (T, U, U , . . . ) into a natural
pseudo-orthonormal one. Firstly, under the transformation
U0 := T , U1 := U, U2 := −U − 1
2 (U · U ) T , (25)
the initial relations (11) transform to
U0 · U0 = 0, U0 · U1 = 0, U0 · U2 = 1,
U1 · U1 = 1, U1 · U2 = 0,
U2 · U2 = 0.
(26)
Secondly, by the standard orthonormalization process we may complete this triple to a tractor frame
(U0, U1, U2; U3, . . . , Un+1) (27)
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 123
such that Ui belongs to the span T , U, . . . , U(i−1)
, for all admissible i, and the corresponding Gram
matrix is
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
1
1
1
...
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (28)
More concretely, for any i = 3, . . . , n, the frame tractors are
Ui = V i/ V i · V i, where
V i = U(i−1)
− (U(i−1)
· Ui−1) Ui−1 − · · ·
· · · − (U(i−1)
· U0) U2 − (U(i−1)
· U1) U1 − (U(i−1)
· U2) U0.
(29)
The last tractor Un+1 is determined by the orthocomplement in T to the codimension one subbundle
T , . . . , U(n−1)
= U0, . . . , Un up to orientation. We do not need to fix the orientation now, it is done
only later. The frame (27) is called the tractor Frenet frame associated to the curve Γ.
Now, differentiating the constituents of the tractor Frenet frame with respect to the conformal arc-length
s and expressing the result within that frame yields Frenet-like identities. As in the classical situation,
it follows that the coefficients of that system determine the generating set of absolute invariants of the
curve. The pseudo-orthonormality of the tractor Frenet frame implies a lot of symmetries among these
coefficients.
Just from (25) we have
U0 = U1, (30)
U1 = K1U0 − U2, where K1 := −1
2 U · U (31)
is the first nontrivial coefficient, the first conformal curvature of the curve. In general, for any admissible i,
the derivative Ui is a linear combination of tractors U0, U1, . . . , Ui+1, namely,
Ui = (Ui · U2) U0 + (Ui · U1) U1 + (Ui · U0) U2 + (Ui · U3) U3 + · · · + (Ui · Ui+1) Ui+1,
where the coefficients satisfy Ui ·Uj = −Ui ·Uj and they have to vanish if i = j or j ≥ i+2. In particular,
from (30) we may read U0 · U1 = 1 and U0 · Uj = 0, for j = 0, 2, . . . , n + 1. Similarly, from (31) we further
read U1 · U2 = K1 and U1 · Uj = 0, for j = 1, 3, . . . , n + 1. Substituting these facts into the previous
display, for i = 2, we see that the only unknown coefficient is the one of U3, i.e. U2 · U3. It however follows
that this coefficient is constant so that we have
Lemma 4.1.
U2 = −K1U1 − U3. (32)
Proof. From (29), according to (25) and (11), we obtain
V 3 = U − (U · U) U + (U · U ) T .
124 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Using U · U = −U · U , which is a consequence of U · U = 0, we obtain
V 3 · V 3 = U · U − (U · U )2
= Φ = 1
and thus U3 = V 3. From (25) and (29) we know that U2 equals to −U mod T , U , respectively
−V 3 mod U0, U1 . Hence U2 · U3 = −V 3 · U3 = −1. 2
Following the same ideas as before, one gradually and easily obtains
U3 = U0 + K2U4,
Ui = −Ki−2Ui−1 + Ki−1Ui+1, for i = 4, . . . , n,
Un+1 = −Kn−1Un,
(33)
where K2, . . . , Kn−1 are the higher conformal curvatures of the curve. The bunch of equations (30), (31),
(32) and (33) form the tractor Frenet equations associated to the curve. Schematically they may be written
as
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U2
U3
U4
...
Un+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
K1 −1
−K1 −1
1 K2
−K2
...
...
Kn−1
−Kn−1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U2
U3
U4
...
Un+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (34)
Altogether, we summarize as follows:
Proposition 4.2. The system of tractor Frenet equations determines a generating set of absolute conformal
invariants of the curve. On a conformal Riemannian manifold of dimension n, it consists of n−1 conformal
curvatures that are expressed, with respect to the conformal arc-length parametrization, as
Ki =
U1 · U2, for i = 1,
Ui+1 · Ui+2, for i = 2, . . . , n − 1.
(35)
In (31) we have an alternative expression of the first conformal curvature in terms of initial tractors.
Thus, as a consequence of Proposition 2.1 and the ensuing definition, we obtain an interpretation of that
invariant:
Proposition 4.3. The conformal arc-length parameter belongs to the projective family of parameters if and
only if K1 = 0.
Remarks 4.4.
(1) The matrix in (34) is skew with respect to the inner product corresponding to (28). A change of basis
leads to a different matrix realization, e.g., just a permutation of the frame tractors leads to following
rewriting of (34):
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 125
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U3
...
Un
Un+1
U2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
K1 −1
1 K2
−K2
...
...
Kn−1
−Kn−1
−K1 −1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U3
...
Un
Un+1
U2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
This block decomposition reflects the standard grading of the Lie algebra so(n+1, 1) with the parabolic
subalgebra corresponding to the block lower triangular matrices. This choice is very close to the ones
in [20] or [5].
(2) For generic curves, the (n + 2)-tuple of derived tractors is linearly independent. For specific curves, this
is not the case and we cannot build the full tractor Frenet frame and all tractor Frenet formulas. One
may however mimic the previous procedure as long as the tractors are independent:
If, say, the tractors T , U, U , . . . , U(i)
are linearly independent and U(i+1)
belongs to their span,
then they form a parallel subbundle in T of rank i + 2 with the corresponding orthonormal tractors
U0, U1, U2, . . . , Ui+1. This yields a part of the tractor Frenet formulas above, in which the first
i−1 conformal curvatures occur; the remaining ones may be considered to be zero. (In this vein, conformal
circles may be considered as the most degenerate curves whose all conformal curvatures vanish, cf.
Proposition 3.3.) In the flat case, this means that the curve itself is contained in an i-sphere, respectively
i-plane, i.e. in the conformal image of an i-dimensional Euclidean subspace. See also [9, section 12] for
further details.
(3) For space- and time-like curves in the case of indefinite signature, we remark the following: The same
transformations as in (25) lead to the relations (26) where the type of curve is reflected just in the sign
of the middle element, hence also in the signature of the subbundle T , U, U in T . The orthogonal
complement of this subbundle in T has in general indefinite signature. Thus, the tractors V i from (29)
may have any sign, including zero. If the sign is negative, we just adjust the corresponding definition
of Ui and, accordingly, one sign in the bottom-right block of (28) changes. If the tractor V i is isotropic
then we cannot satisfy both the diagonal form of the bottom-right block of (28) and the condition
Ui ∈ T , U, . . . , U(i−1)
. Of course, one can always transform given tractors to a pseudo-orthonormal
tractor frame, but only with some additional choices. The rest remains basically the same, only the
tractor Frenet formulas may look different and some of the resulting invariants has to be related to
the respective choices. (Nevertheless, it is clear that the exceptional conformal curvature K1 and its
interpretation as in Proposition 4.3 are not influenced by these issues.) These observations lead to a finer
type characterization of space- and time-like curves. Its complete branching structure is manageable
only in a concrete dimension and signature.
4.2. Conformal curvatures revised
In the definition above, the orientation of the last tractor from the tractor Frenet frame was a matter of
choice. In what follows we assume the orientation of Un+1 is chosen so that it belongs to the same half-space
as U(n)
with respect to the hyperplane U0, . . . , Un . Thus, according to (29), we have
U(i)
= V i+1 mod U0, U1, . . . , Ui , (36)
126 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
for any i = 3, . . . , n. Also, from definitions and the tractor Frenet formulas, one progressively obtains
T = U0, U = U1, U = −U2 + K1U0, U = U3 + 2K1U1 + K1U0, etc. (37)
In particular, it holds U = U3 mod U0, U1, U2 . Similarly, it follows that U = K2U4 mod
U0, U1, U2, U3 and, inductively,
U(i)
= K2 · · · Ki−1Ui+1 mod U0, U1, . . . , Ui , (38)
for i = 3, . . . , n. Altogether, from (36), (38) and Ui = V i/
√
V i · V i, we obtain
K2 · · · Ki−1 = V i+1 · V i+1,
which leads to the following conclusion:
Proposition 4.5. With respect to the conformal arc-length parametrization, the higher conformal curvatures
can be expressed as
Ki =
V i+2 · V i+2
V i+1 · V i+1
, for i = 2, . . . , n − 1. (39)
In particular, all these curvatures are positive. The first conformal curvature K1 does not fit to this
uniform description and its sign may be arbitrary, including zero.
An expansion of (39), according to (25) and (29), provides expressions of higher conformal curvatures in
terms of initial tractors. Instead of exploiting this demanding procedure, we are going to use the relative
conformal invariants from subsection 2.3. From the construction of the tractor Frenet frame it follows
that Δi+1 = Δi(V i · V i), for all admissible i. Since Δ3 = −1, we see that all admissible determinants
are negative. Hence we can express V i · V i as a quotient of determinants, whose substitution into (39)
yields
Ki =
Δi+1Δi+3
−Δi+2
. (40)
Up to this point, everything has been related to the parametrization by the conformal arc-length. Now
we consider an arbitrary parametrization of the curve.
Theorem 4.6. With respect to an arbitrary parametrization of the curve, the higher conformal curvatures
can be expressed as
Ki =
Δi+1Δi+3
−Δi+2
4
√
−Δ4
, for i = 2, . . . , n − 1. (41)
Proof. According to Lemma 2.2, the weight of the right hand side of (41) is
1
2 (i + 1)(i − 2) + 1
2 (i + 3)i − (i + 2)(i − 1) − 1 = 0,
thus it defines an absolute conformal invariant. With respect to the conformal arc-length parametrization,
(41) coincides with (40), hence the statement follows. 2
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 127
Alternatively, the statement is deducible from (40) by the reparametrization according to subsection 2.2
and Lemma 2.2, where g = Φ
1
4 . That way, we may obtain also an expression of the very first conformal
invariant, K1, which does not fit to the just given description: Its expression with respect to the conformal
arc-length is given in (31), which transforms under reparametrizations according to (15). The substitution
of g = Φ
1
4 and a small computation reveals that
Proposition 4.7. With respect to an arbitrary parametrization of the curve, the first conformal curvature can
be expressed as
K1 = −1
2 Φ− 1
2 U · U − 1
2 (ln Φ) + 1
16 (ln Φ) 2
=
= −1
2 Φ− 5
2 Φ2
U · U − 1
2 ΦΦ + 9
16 Φ 2
.
(42)
Remarks 4.8.
(1) Not only the conformal curvatures, but the tractor Frenet frame itself can be built with respect to
an arbitrary parametrization. E.g. expressions of the first three tractors (25) transform, under the
reparametrization according to (14) with g = Φ
1
4 , so that
U0 = Φ
1
4 T ,
U1 = U + 1
4 (ln Φ) T ,
U2 = −Φ− 1
4 U + 1
4 (ln Φ) U + 1
2 U · U + 1
32 (ln Φ) 2
T .
(43)
(2) Our exceptional invariant K1 corresponds to a similarly exceptional invariant Jn−1 of [9, section 5] as
follows. For any relative conformal invariant Q of weight 1, an appropriate combination of Q and its
derivatives leads to an appropriate transformation of the new quantity under reparametrizations of the
curve. Namely, with the same conventions as yet,
2QQ − 3Q 2
= g −4
2QQ − 3Q 2
− 2S(g)Q2
.
Combining with (15), it easily follows that the function
Q−2
U · U − 2Q−3
Q + 3Q−4
Q 2
is an absolute conformal invariant of the curve. Now, the substitution Q = Φ
1
4 leads to an invariant,
which differs from our K1, respectively Fialkow’s Jn−1, just by a constant multiple. (For the comparison
with Jn−1, the formula (44) from the next subsection is needed.)
4.3. Conformal curvatures in terms of Riemannian ones
Expanding any of the above expressions for the conformal curvatures yields very concrete and very ugly
formulas in terms of the underlying derived vectors. As a compromise between the explicitness and the
ugliness we show how to rewrite the previous formulas in terms of the Riemannian curvatures of the curve.
This way we will also be able to compare our invariants with their more classical counterparts.
Within this paragraph we consider a generic curve parametrized by the Riemannian arc-length parameter,
according to a chosen scale. The corresponding Riemannian Frenet frame is denoted by (e1, . . . , en) and the
Riemannian Frenet formulas are2
2
Warning: in this subsection we relax the abstract indices, i.e. we write ei rather than ea
i etc. The raising and lowering indices
with respect to the chosen metric is denoted by and , respectively.
128 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
e1 = κ1e2,
ei = −κi−1ei−1 + κiei+1, for i = 2, . . . , n − 1,
en = −κn−1en−1,
where primes denote the derivative with respect to the Riemannian arc-length and κ1, . . . , κn−1 are the
Riemannian curvatures of the curve. Accordingly, the expressions (3) and (4) read as
u = 1, U = e1, U = κ1e2, U = −κ2
1e1 + κ1e2 + κ1κ2e3, etc.
Following the development of subsection 2.1, the first tractors are
T =
0
0
1
, U =
0
e1
0
, U =
−1
κ1e2
−P(e1, e1)
,
U =
0
−κ2
1e1 + κ1e2 + κ1κ2e3 − P(e1, e1)e1 − P(e1, −)
∗
.
Hence we obtain
U · U = κ2
1 + 2P(e1, e1),
Φ = U · U − (U · U )2
= κ 2
1 + κ2
1κ2
2 + terms involving P.
(44)
Substitution of (44) into (42) leads to an expression of the first conformal curvature K1 in terms of first
two Riemannian curvatures κ1, κ2, some terms involving the Schouten tensor and their derivatives. The full
expansion leads to a huge formula even in the flat case.
Formula enthusiasts may continue further in this spirit and express the higher derived tractors. We add
some details on the Gram matrix corresponding to the first five tractors. It is displayed in (13) where,
modulo terms involving P,
α = κ2
1, β = κ 2
1 + κ4
1 + κ2
1κ2
2,
γ = 9κ2
1κ 2
1 + (κ1 − κ3
1 − κ1κ2
2)2
+ (2κ1κ2 + κ1κ2)2
+ κ2
1κ2
2κ2
3.
Of course, the term κ3 is nontrivial only if n ≥ 4. From this and (41) one could deduce an expression for
the second conformal curvature K2 in terms of κ1, κ2, κ3, P and their derivatives. Although the result is
expected to be messy, it can be condensed into the following neat form:
K2 = κ 2
1 + κ2
1κ2
2
− 5
4
(2κ 2
1 κ2 + κ2
1κ3
2 + κ1κ1κ2 − κ1κ1 κ2)2
+ (κ 2
1 + κ2
1κ2
2)κ2
1κ2
2κ2
3
1
2
,
modulo terms involving P.
Remark 4.9. In the flat case, for n = 3, the previous display agrees precisely with the invariant which is
called conformal torsion in [5] (the other invariant there corresponds to our K1). Note that vanishing of K2
in such case is equivalent to κ2 = 0, or κ2 = 0 and κ2
κ1
=
κ1
κ2
1κ2
. The former condition means that the
curve is planar, the latter one means the curve is spherical. By virtue of Remark 4.4(2), this is an expected
behaviour.
For another comparisons let us also describe first few tractors from the tractor Frenet frame constructed
in subsection 4.1. According to (43), the easy tractors are
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 129
U0 = Φ
1
4
0
0
1
, U1 =
⎛
⎝
0
e1
1
4 (ln Φ)
⎞
⎠ , U2 = −Φ− 1
4
⎛
⎜
⎝
−1
1
4 (ln Φ) e1 + κ1e2
1
32 (ln Φ) 2
+ 1
2 κ2
1
⎞
⎟
⎠ ,
where Φ is given in (44). The first nontrivial step concerns an expression of U3, which one calculates as
U3 = Φ− 1
2
⎛
⎝
0
κ1e2 + κ1κ2e3 + P(e1, e1)e1 − P(e1, −)
κ1κ1
⎞
⎠ . (45)
Remarks 4.10.
(1) Incidentally, the middle slot of the tractor in (45) corresponds to the starting quantity in Fialkow’s
approach, the so-called first conformal normal of the curve, cf. [9, equation (4.21)]. Using this and an
invariantly defined derivative along the curve (with respect to the conformal arc-length parameter), the
conformal Frenet identities and the corresponding n − 2 conformal curvatures are deduced in [9, section
4]. The last Fialkow’s conformal curvature is constructed ad hoc in [9, section 5], see the discussion
in Remark 4.8(2).
(2) At this stage we can easily count the orders of individual invariants although some estimates could be
done earlier. The relative conformal invariant Φ is of order 3, cf. (44). From this and (42) we see that
the first conformal curvature K1 is of order 5. For i ≥ 2, an estimate based on (41) yields the order of
ith conformal curvature Ki is at most i + 3, but the current expressions show it is actually i + 2.
5. Null curves in general signature
The next step is to consider possible generalizations of the previous treatment to conformal manifolds
of arbitrary signature. Thus, in the following we suppose that M is a conformal manifold of dimension
n = p + q and signature (p, q) and T is the standard tractor bundle with the tractor metric of signature
(p + 1, q + 1). Generally, we should distinguish curves according to the type of tangent vectors/tractors
and also according to the type of further tractors in the tractor Frenet frame. This leads to a diversity of
possible cases, cf. Remark 4.4(3). The typical situation we need to understand is when the tangent vector
together with its several derivatives generate a totally isotropic subspace of the tangent space, cf. the notion
of r-null curves below. One can view this as a counterpart of the Riemannian setting discussed so far. In
the last subsection we discuss the Lorentzian signature in more detail.
5.1. r-null curves
For null curves the density (3) vanishes identically and cannot be used for the lift to T . Depending on
the signature, we may consider curves that are more and more isotropic, without being degenerate. On a
conformal manifold of signature (p, q), for any r ≤ min{p, q}, a curve is called the r-null curve if the vectors
Ua
, . . . , U(r−1)a
are linearly independent and null, whereas U(r)a
is not null. Consequently, it holds
U(i)
c U(j)c
=
0, for i + j < 2r,
(−1)r−i
U
(r)
c U(r)c
, for i + j = 2r.
(46)
Note that the notion of r-null curves is well defined as it does not depend on the chosen scale. Indeed, one
can easily verify by induction using (1) that
U(i)a
= U(i)a
mod Ua
, . . . , U(i−1)a
,
130 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
for all 1 ≤ i ≤ 2r. In this vein, space- and time-like curves may be termed 0-null curves. As an opposite
extreme, we use the notation ∞-null curves for curves, whose all vectors U(i)a
are isotropic; such curves
are necessarily degenerate. The most prominent—and the most degenerate—instance of such curves are the
null geodesics.
For an r-null curve, it follows from (1) that the norm squared of U(r)c
is a nowhere vanishing density of
conformal weight 2. Thus,
u := |U
(r)
c U(r)c| ∈ E[1]
provides a lift T ∈ T as in (7), to which we add the derived tractors U, U etc. as before. Explicit expressions
are very analogous to those in (9), (10) etc. up to some shift. For i ≤ 2r, we have
U(i)
=
⎛
⎜
⎝
0
u−1
U(i)a
+ · · ·
∗
⎞
⎟
⎠ ,
where the zero in the primary slot appears as −u−1
UcU(i−1)c
= 0 and the dots in the middle slot denote
terms of lower order. In particular, we have T · T = · · · = U(r−1)
· U(r−1)
= 0 and
:= U(r)
· U(r)
= ±1. (47)
The first nontrivial quantity is U(r+1)
· U(r+1)
. In order to simplify expressions, we will use the notation
α := U(r+1)
· U(r+1)
, β := U(r+2)
· U(r+2)
, γ := U(r+3)
· U(r+3)
, etc. (48)
The pattern of the initial relations remains the same as in (13) up to the signs concerning and a shift in
the south-east direction. After some computation, one shows that the first few antidiagonals of the Gram
matrix read as
U(i)
· U(j)
=
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0, for i + j < 2r,
(−1)r−i
, for i + j = 2r,
0, for i + j = 2r + 1,
(−1)r+1−i
α, for i + j = 2r + 2,
(−1)r+1−i 2(r−i)+3
2 α , for i + j = 2r + 3,
(−1)r−i
β + (−1)r+1−i (r−i+2)2
2 α , for i + j = 2r + 4,
(49)
where we use the convention U(−1)
:= T .
Under the reparametrization ˜t = g(t) as in subsection 2.2, one easily verifies that T = g r+1
T , from
which one deduces the transformations of higher order tractors. Considering U(i)
:= 0, for i < −1, it turns
out that
U
(k)
= g r−k
U(k)
+ Akg −1
g U(k−1)
+ (Bkg −1
g − Ckg −2
g 2
)U(k−2)
+ · · · , (50)
for k = 0, 1, . . . , where the coefficients Ak, Bk, Ck are given by the recurrence relations
Ak+1 = Ak + r − k,
Bk+1 = Bk + Ak,
Ck+1 = Ck − (r − k − 1)Ak,
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 131
with the initial conditions A0 = r + 1, B0 = 0 and C0 = 0. It is an elementary, but tedious, exercise to solve
and tidy up this system. For k = r + 1, it turns out that
Ar+1 = 1
2 (r + 1)(r + 2),
Br+1 = 1
6 (r + 1)(r + 2)(2r + 3),
Cr+1 = −1
8 (r + 1)(r + 2)(r2
− r − 4).
From this and the initial relations it follows that
α = g −2
α − 2 Br+1g −1
g + (A2
r+1 + 2Cr+1)g −2
g 2
.
Since A2
r+1 + 2Cr+1 = 3Br+1, the previous simplifies to
α = g −2
(α − 2 Br+1S(g)) . (51)
Thus, the equation α = 0, regarded as a condition on the parametrization of an r-null curve, determines a
preferred family of parametrizations with freedom the projective group of the line. This is a generalization
of Proposition 2.1 which just corresponds to r = 0. Altogether we may conclude with
Proposition 5.1. For any admissible r, any r-null curve carries a preferred family of projective parameters.
It is well known that null geodesics carry a preferred family of affine parameters.
Further, the Gram determinants are defined and denoted just the same as in (17). Initial relations for
an r-null curve lead to Δi = 0, for i = 1, . . . , 2r + 2, and Δ2r+3 = − . The first potentially nontrivial
determinant is Δ2r+4. Analogously to Lemma 2.2 we have
Lemma 5.2. Along any r-null curve, the Gram determinant Δi, for i = 2r + 4, . . . , n + 2, is a relative
conformal invariant of weight i(i − 2r − 3).
For any i = 2r + 4, . . . , n + 2, the vanishing of Δi is equivalent to the fact that the determining tractors
are linearly dependent. The weight of any Δi is even, hence its sign is not changed by reparametrizations. In
contrast to Lemma 2.3 (corresponding to r = 0), we cannot say anything about the particular sign of Δ2r+4.
For an r-null curve with nowhere vanishing Δ2r+4, we define the conformal pseudo-arc-length parameter by
integration of the 1-form
ds := 2r+4
|Δ2r+4(t)| dt (52)
along the curve; it is given uniquely up to an additive constant. The invariant Δ2r+4 (hence also all higher
ones) may vanish. In particular, this happens automatically if 2r + 4 > n + 2, i.e. if n < 2r + 2. From the
introductory counts we know that r and n are related by n ≥ 2r, therefore the critical dimensions are n = 2r
and 2r + 1. In such cases, none of the just considered invariants can be used to define a natural parameter
of the curve. However, there is another couple of relative conformal invariants as shown in subsection 5.2
below; see also Remark 5.15 for further comments.
At this stage, we sketch how to adapt the construction of the tractor Frenet frame from subsection 4.1
for r-null curves with the pseudo-arc-length defined. The smallest nondegenerate subbundle in T gradually
built from the derived tractors is T , U, . . . , U(2r+1)
. The restriction of the tractor metric to this subbundle
132 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
has signature (r + 2, r + 1), respectively (r + 1, r + 2), thus its orthogonal complement in T has signature
(p − r − 1, q − r), respectively (p − r, q − r − 1). We may always start with the prescription
U0 := T , U1 := U, . . . , Ur+1 := U(r)
. (53)
Then we have to consider transformations determining Ui ∈ T , U, . . . , U(i−1)
, for i = r + 2, . . . , 2r + 2,
so that the Gram matrix corresponding to the tractors (U0, . . . , U2r+2) is antidiagonal with values ±1
(this substitutes the upper-left block in (28)). Note that in contrast to the case r = 0, the definition for
Ur+2, . . . , U2r+2 is not unique. This freedom is already visible from the first tractor of this subsequence
which may be given as
Ur+2 = − U(r+1)
− 1
2 αU(r−1)
+ · · · , (54)
where the dots stay for any linear combination of lower order tractors U(r−2)
, . . . , T . To accomplish the full
tractor frame, i.e. to determine (U2r+3, . . . , Un+1), we continue by an orthonormalization process. Going
along, we may meet an isotropic tractor: in such case we face the same problem as discussed in Remark 4.4(3)
and some additional choices are inevitable.
Having constructed the tractor Frenet frame, we differentiate with respect to the conformal pseudo-arclength
to obtain the tractor Frenet equations with the generating set of absolute conformal invariants. Just
from (53) and (54) we easily deduce first few Frenet-like identities:
U0 = U1, . . . , Ur = Ur+1,
Ur+1 = K1Ur − Ur+2, where K1 := −1
2 α.
Clearly, the first conformal curvature is not affected by the freedom in the construction of the tractor
Frenet frame. From the expression of K1 and the remark preceding Proposition 5.1 we immediately have
the following generalization of Proposition 4.3.
Proposition 5.3. The conformal pseudo-arc-length parameter belongs to the projective family of parameters
of an r-null curve if and only if K1 = 0.
The freedom in the construction of (Ur+2, . . . , U2r+2) influences the corresponding part of tractor
Frenet equations and so the corresponding conformal curvatures, namely, K2, . . . , Kr+1. The respective
freedom in the construction of (U2r+3, . . . , Un+1) influences some of the remaining conformal curvatures
Kr+2, . . . , Kn−r−1. Notably, they are n − r − 1 in total.
Various expressions analogous to (35) and (39) (with respect to the conformal pseudo-arc-length parameter),
or (41) and (42) (with respect to an arbitrary parameter) are deducible. We provide more details only
in subsection 5.4 for the case of Lorentzian signature.
5.2. Invariants of Wilczynski type
For an r-null curve, vanishing of Δ2r+4 means that the tractors T , U, . . . , U(2r+2)
are linearly dependent.
Since the first 2r +3 tractors from this sequence are linearly independent and U(k)
= T (k+1)
, this condition
may be interpreted as a tractor linear ODE,
T (2r+3)
+ q2r+2T (2r+2)
+ · · · + q1T + q0T = 0, (55)
where q0, . . . , q2r+2 are functions expressible in terms of pairings of the initial tractors. Following [21, ch. II,
§4], we may consider the so-called Wilczynski invariants of this equation, i.e. the set of 2r + 1 essential
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 133
invariants denoted as Θ3, . . . , Θ2r+3 (the indices indicate weights). Of course, we may associate Wilczynski
invariants to the linear differential operator on the left hand side of (55), i.e. also to general r-null curves
with nonzero Δ2r+4. As a matter of fact, these invariants are relative conformal invariants of the curve.
Assuming q2r+2 = 0, i.e. the so-called semi-canonical form, an explicit form of first two of these invariants
is given by the following Lemma.
Lemma 5.4 ([17]). Let y(k)
+ qk−2y(k−2)
+ · · · + q0y = 0 be a linear ODE of kth order in the semi-canonical
form (qk−1 = 0). Then the first two Wilczynski invariants are
Θ3 = qk−3 −
k − 2
2
qk−2,
Θ4 = qk−4 −
k − 3
2
qk−3 +
(k − 2)(k − 3)
10
qk−2 −
(5k + 7)(k − 2)(k − 3)
10k(k + 1)(k − 1)
q2
k−2.
The formula for Θ3 and Θ4 was known already to Laguerre and Schlesinger, respectively, both predating
the work of Wilczynski. Accurate references and modern treatment of this classical subject can be found in
[18, section 4.1].
Now we need to relate coefficients qi from the equation (55) to quantities α, β, γ etc. from (48). By pairing
both sides of (55) with T , it follows from (49) that q2r+2 = 0, i.e. the equation is in the semi-canonical
form. Considering similarly the pairing of both sides of (55) with T = U, T = U and with T = U ,
one computes
q2r+2 = 0, q2r+1 = α, q2r =
2r + 1
2
α , q2r−1 = − β +
r2
2
α + α2
. (56)
Combining this with Lemma 5.4, we obtain the resulting form of Θ4:
Proposition 5.5. Let , α and β be the quantities associated to an r-null curves (with respect to an arbitrary
parametrization) as in (47) and (48). Then Θ3 = Θ5 = 0 and
Θ4 = − β −
r(r + 3)
10
α +
(r + 3)(2r + 5)(5r + 4)
10(r + 1)(r + 2)(2r + 3)
α2
is a relative conformal invariant of the weight 4.
Proof. Both vanishing of Θ3 and the form of Θ4 follows from (56) and Lemma 5.4, where k = 2r + 3. The
particular weight of Θ4 follows from the definitions (48) and the exponent of g in (50).
To show Θ5 = 0, we shall for simplicity assume a projective parametrization of the curve, i.e. α = 0. In
particular, the equation is in the canonical Laguerre–Forsyth form for which the expressions of Wilczynski
invariants are well known, see [21, ch. II, eqn. (48)]. Analogously to the computation leading to (56), we
obtain
U(i)
· U(2r+5−i)
= (−1)r−i 2(r − i) + 5
2
β and q2r−2 = −
2r − 1
2
β .
Now substituting coefficients q2r−1 and q2r−2 into the just referred formula, one easily verifies that
Θ5 = 0. 2
Observe that Θ4 recovers Δ4 from (18) for r = 0.
134 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
Remarks 5.6.
(1) The Laguerre–Forsyth form of the equation is preserved by transformations with freedom the projective
group of the line, which corresponds precisely to the condition α = 0. Also, it follows that the linear
equation is equivalent to the trivial equation if and only if all Wilczynski invariants vanish. In our case,
this is equivalent to the vanishing of α, β, etc. from (48) up to U(2r+2)
· U(2r+2)
.
(2) Experiments with specific r’s suggest the equation (55) could be self-adjoint, i.e. all odd Wilczynski
invariants vanish. This would mean that we have just r (rather than 2r) new and potentially nontrivial
invariants Θ4, . . . , Θ2r+2.
5.3. Conformal null helices
For any space- or time-like curve with an arbitrary parametrization, a metric in the conformal class may
be chosen so that U a
= 0, cf. (5). For null curves, this condition holds only if the curve is null geodesic, i.e.
the null curve whose acceleration vector U a
is proportional to Ua
. Consequently, all higher order vectors
are proportional to Ua
. These curves, although very important, cannot be lifted to the standard tractor
bundle in the sense considered above. That is why they are excluded from our considerations.
For an r-null curve, we see from (46) and the subsequent discussion that the vectors Ua
, . . . , U(2r)a
are
linearly independent for any choice of scale. Therefore we cannot achieve any of the conditions U a
= 0,
. . . , U(2r)a
= 0 by a conformal change of metric. The simplest conceivable statement is
Lemma 5.7. For an r-null curve with an arbitrary parametrization, a metric in the conformal class may be
chosen so that U
(r)
c U(r)c
= ±1 and U(2r+1)a
= 0.
Proof. This can be proved by various means. For the sake of latter references, we employ the tractors
anyhow it may seem artificial. Firstly, we may always choose the metric so that U
(r)
c U(r)c
= ±1. With this
assumption, the tractors associated to the curve have the form
T =
0
0
1
, U =
0
Ua
0
, U =
0
U a
ρ1
, . . . ,
U(j)
=
⎛
⎝
εj
U(j)a
mod Ua
, . . . , U(j−2)a
ρj
⎞
⎠ ,
for j = 2, . . . , 2r + 1, where ρj are in general nonzero while all εj vanish except for ε2r+1 = (−1)r−1
.
Secondly, the choice of metric may be adjusted so that the bottom slot of U vanishes and the expressions
of previous tractors are unchanged: according to (6), the corresponding Υa has to satisfy ΥcU c
= ρ1 and
ΥcUc
= 0. Hence U =
0
U a
ρ2
, i.e. there is only the leading term in the middle slot. Inductively, we may
achieve a rescaling so that U(j)
=
0
U(j)a
0
, for all j ≤ 2r. Hence U(2r+1)
=
(−1)r−1
U(2r+1)a
∗
. Finally, we may
consider a rescaling so that the middle slot of the last tractor vanishes: according to (6), the corresponding
Υa
equals to (−1)r
U(2r+1)a
along the curve. Hence the statement follows. 2
Now we are ready to identify the closest relatives of conformal circles among r-null curves. We know
from preceding subsections that the first 2r +3 derived tractors T , U, . . . , U(2r+1)
are linearly independent.
Therefore we can never achieve any of the conditions U = 0, . . . , U(2r+1)
= 0 for r-null curves. The
simplest conceivable condition appears in item (a) of the following Theorem.
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 135
Theorem 5.8. For an r-null curve, the following conditions are equivalent:
(a) the curve with a projective parametrization (α = 0) obeys U(2r+2)
= 0,
(b) the curve (with and arbitrary parametrization) obeys Δ2r+4 = 0 and Θ4 = · · · = Θ2r+3 = 0,
(c) there is a metric in the conformal class such that U
(r)
c U(r)c
= ±1, U(2r+1)a
= 0 and Uc
Pca = 0.
Proof. Expressing the condition Δ2r+4 = 0 as (55), the equivalence of (a) and (b) follows from the general
theory of linear differential equations and the fact that Θ3 vanishes automatically, cf. Proposition 5.5 and
Remark 5.6(1).
For the rest we assume the scale is chosen as in the proof of Lemma 5.7. In such scale, the derived tractors
are
T =
0
0
1
, U =
0
Ua
0
, . . . , U(2r)
=
⎛
⎝
0
U(2r)a
0
⎞
⎠ , U(2r+1)
=
⎛
⎝
(−1)r−1
0
−Uc
U(2r)d
Pcd
⎞
⎠ .
The additional assumption Uc
Pca = 0 from (c) then yields U(2r+2)
= 0 and so (a) holds.
Conversely, from (a) it in particular follows U(2r+1)
· U(2r+1)
= 0, hence the bottom slot of U(2r+1)
has
to vanish. Then U(2r+2)
= 0 implies that Uc
Pca = 0 and so (c) holds. 2
Any curve satisfying (any of) the conditions (a)–(c) is called conformal r-null helix. The condition
Δ2r+4 = 0 is equivalent to the fact that the subbundle T , U, . . . , U(2r+1)
⊂ T of rank 2r + 3 is parallel
along the curve. In the flat case, this means that the conformal r-null helix is contained in the conformal
image of a (2r + 1)-dimensional Euclidean subspace. With condition (c), it easily follows that these curves
are just the conformal images of so-called null Cartan helices, cf. [7]. Note that the conformal circles fit to
the just given description for r = 0.
Remark 5.9. As in subsection 3.3, one may construct conserved quantities along conformal r-null helices
on specific conformal manifolds. The two examples mentioned in Propositions 3.5 and 3.6 have obvious
counterparts for general r, provided that the respective functions are replaced by
s := U(2r+1)
· S and k := K(U, U(2r+1)
).
The reasoning is the same, explicit expressions are analogous, but more complicated.
5.4. Remarks on Lorentzian signature
In this subsection we suppose an n-dimensional conformal manifold of Lorentzian signature (n − 1, 1),
hence the standard tractor bundle T with the bundle metric of signature (n, 2). Following the scheme of
previous subsections, a lot of things simplify as the maximal isotropic subspace in the tangent space has
dimension one.
The only admissible r-null curves in this signature correspond to r = 1. Therefore we may speak without
a risk of confusion just about null curves, instead of 1-null curves. The assumption UcUc
= 0 implies that
UcU c
= 0, thus the vector U a
must be space-like. The lift to T is provided by the density u = UcU c. The
smallest nondegenerate subbundle in T built from the derived tractors is of rank m5 and has signature (3, 2).
Hence its orthogonal complement is positive definite. The first few starting relations are indicated in the
following Gram matrix for the sequence T , U, U , U , U , U , cf. (48) and (49):
136 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 0 0 0 1 0
0 0 0 −1 0 α
0 0 1 0 −α −3
2 α
0 −1 0 α 1
2 α 1
2 α − β
1 0 −α 1
2 α β 1
2 β
0 α −3
2 α 1
2 α − β 1
2 β γ
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
The projective parametrization of the null curve corresponds to α = U · U = 0.
The trivial Gram determinants are Δ1 = Δ2 = Δ3 = Δ4 = 0 and Δ5 = 1. The first nontrivial one is
Δ6 = γ + α α − 2βα − 9
4 α 2
+ α3
,
provided that the dimension of conformal manifold is n ≥ 4, which is assumed in the rest of this subsection
(the special case n = 3 is dealt individually in Remark 5.15). This invariant is nonnegative:
Lemma 5.10. Δ6 ≥ 0.
Proof. For a projective parametrization of the null curve, α = 0, we have Δ6 = γ. According to this choice,
one easily verifies that the tractor
V 5 := U − β U − 1
2 β T
belongs to the orthogonal complement of the subbundle T , U, . . . , U in T . This complement has the
positive definite signature, hence we conclude with γ = U · U = V 5 · V 5 ≥ 0. Since the weight of Δ6
is even, it is a nonnegative function independently of the parametrization of the curve. 2
Accordingly, the 1-form (52) leading to the definition of conformal pseudo-arc-length may be substituted
by
ds := 6
Δ6(t) dt. (57)
Clearly, vanishing of Δ6 is equivalent to vanishing of V 5, which yields exactly the equation (55), for
r = 1. Among the corresponding Wilczynski invariants, there is only one nontrivial, namely,
Θ4 = − 1
25 (25β + 10α − 21α2
), (58)
cf. Proposition 5.5. According to Theorem 5.8, we may conclude with
Proposition 5.11. Conformal null helices are characterized by the pair of equations
Δ6 = 0 and Θ4 = 0.
To build the tractor Frenet frame, we start with
U0 := T , U1 := U, U2 := U . (59)
Now, all admissible transformations determining U3 and U4, so that the 5-tuple (U0, . . . U4) has the
antidiagonal Gram matrix, are
U3 := −U − 1
2 α U + k T ,
U4 := U + α U + U + 1
2 (α2
− β) T ,
(60)
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 137
where k + = 1
2 α . For the rest of this demonstration we choose
k = 0 and = 1
2 α . (61)
As mentioned above, the orthocomplement to the subbundle T , . . . , U = U0, . . . , U4 has positive
definite signature. Therefore, we easily complete the tractor Frenet frame
(U0, . . . , U4; U5, . . . , Un+1)
in the spirit of (29) so that Ui ∈ T , U, . . . , U(i−1)
, for all admissible i, and the corresponding Gram
matrix is
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
...
1
1
...
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
Now, differentiating with respect to the conformal pseudo-arc-length we approach the identities with the
generating set of absolute conformal invariants. Just from (59) and (60), with respect to the choice (61), we
easily deduce first four Frenet-like identities:
U0 = U1, (62)
U1 = U2, (63)
U2 = K1U1 − U3, where K1 := −1
2 α, (64)
U3 = K2U0 − K1U2 − U4, where K2 := 1
2 (α2
− β). (65)
The next step is subtle, but fully analogous to the one announced in Lemma 4.1: the only coefficient in the
expression of U4, which cannot be deduced from the previous identities, is the one of U5, i.e. U4 · U5. It
however follows that this coefficient is constant, namely,
U4 = −K2U1 + U5. (66)
(The crucial point in the argument is the expression of V 5 and the observation that V 5 · V 5 = Δ6. Since
we assume parametrization by the conformal pseudo-arc-length, this equals to 1, thus U5 = V 5.) The rest
is easy so that one quickly summarizes as
U5 = −U0 + K3U6,
Ui = −Ki−3Ui−1 + Ki−2Ui+1, for i = 6, . . . , n,
Un+1 = −Kn−2Un.
(67)
The cluster of equations from (62) through (66) to (67) forms the tractor Frenet equations associated to the
null curve determining its conformal curvatures. Schematically, the tractor Frenet equations may be written
as
138 J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U2
U3
U4
U5
U6
...
Un+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
1
K1 −1
K2 −K1 −1
−K2 1
−1 K3
−K3
...
...
Kn−2
−Kn−2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
U0
U1
U2
U3
U4
U5
U6
...
Un+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Altogether, we summarize as follows:
Proposition 5.12. The system of tractor Frenet equations determines a generating set of absolute conformal
invariants of the null curve. On a conformal manifold of Lorentzian signature and dimension n ≥ 4, it
consists of n − 2 conformal curvatures that are expressed, with respect to the conformal pseudo-arc-length
parametrization, as
Ki =
⎧
⎨
⎩
Ui+1 · Ui+2, for i = 1, 2
Ui+2 · Ui+3, for i = 3, . . . , n − 2.
Among these curvatures, only K2 depends on additional choices in the construction of U3 and U4.
As in subsection 4.2, there are alternative ways of expressing the conformal curvatures. The first two, i.e.
the two exceptional, conformal curvatures K1 and K2 are given in terms of initial tractors with respect to the
conformal pseudo-arc-length parametrization in (64) and (65). One may obtain the expressions with respect
to an arbitrary reparametrization by a substitution according to (51), respectively (50), where g = 6
√
Δ6.
In the former case, one ends up with the formula very similar to the one in (42), the later case is more
ugly. For the higher conformal curvatures we may proceed as follows. Firstly, an analogue of Proposition 4.5
turns out to be
Proposition 5.13. With respect to the conformal arc-length parametrization, the higher conformal curvatures
can be expressed as
Ki =
V i+3 · V i+3
V i+2 · V i+2
, for i = 3, . . . , n − 2. (68)
Secondly, it holds Δi+1 = Δi(V i · V i), for all admissible i. Since Δ5 = 1, we in particular see that all
admissible determinants are positive. Now, expressing V i · V i via the determinants, substituting into (68)
and passing to an arbitrary parametrization, we obtain the following substitute of Theorem 4.6:
Theorem 5.14. With respect to an arbitrary parametrization, the higher conformal curvatures can be expressed
as
Ki =
Δi+2Δi+4
Δi+3
6
√
Δ6
, for i = 3, . . . , n − 2.
J. Šilhan, V. Žádník / J. Math. Anal. Appl. 473 (2019) 112–140 139
Remark 5.15. As we already noticed, the study of null curves in dimension n = 3 requires an extra care.
The relative invariant Δ6 vanishes automatically in that case, therefore it cannot be used to define the
distinguished parametrization of the curve. However, we still have the Wilczynski invariant Θ4 given by (58).
Thus, for a generic null curve in 3-dimensional manifold, we define an alternative conformal pseudo-arc-length
parameter by integrating the 1-form
ds := 4
|Θ4(t)| dt,
instead of (57). Accordingly, we may follow the construction of conformal invariants as above with the
following conclusion: the whole tractor Frenet frame consists just from the first five tractors displayed in
(59)–(60) and the corresponding tractor Frenet equations are just (62)–(66), with U5 = 0. In those equations,
two conformal invariants appear, namely, K1 and K2. It however follows from their expressions, and the
fact that Θ4 = 1 for the current conformal pseudo-arc-length parametrization, that K2 is expressible as a
function of K1 and its derivatives, namely,
K2 = −
2
5
K1 +
8
25
K2
1 +
1
2
.
Therefore we end up with only one significant absolute conformal invariant, K1, as expected. An interpretation
of that invariant is still the same as in Propositions 5.3.
In the case that Θ4 vanishes identically, we recover the conformal null helices discussed above, cf. Proposition
5.11.
Acknowledgments
Authors thank to Boris Doubrov, Rod Gover and Igor Zelenko for useful discussions. Experiments, checks
and comparisons of some explicit expressions were done with the computational system Maple. JŠ was
supported by the Czech Science Foundation (GAČR) under grant P201/12/G028, VŽ was supported by the
same foundation under grant GA17-01171S.
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