RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO
Serie II, Suppl. 75 (2005), pp. 347-356
REMARKS ON DEVELOPMENT OF CURVES
VOJTECH ZADNIK
ABSTRACT. The classical notion of development of curves easily generalizes to the
wide class of Cartan geometries. This construction provides certain correspondence
between smooth curves on the base manifold and smooth curves in the homogeneous
space of the geometry in question. In this vein, there are specific curves on the base
manifold distinguished in terms of their developments. Explicit relationship between
the initial curve and its development is worked out.
1. INTRODUCTION
Given a smooth manifold M endowed with some geometric structure, certain specific
curves on M become useful in number of problems, since they are very simple objects
which can inherit a considerable amount of the entire geometric information. Typically,
geodesies on Riemannian manifolds are of this feature. This is why distinguished types
of curves appear in other geometries, in some of which they are very well-studied, e.g.,
conformal geodesies and null-geodesies on conformal manifolds. In the framework of
Cartan geometries, these types of curves can be studied in an unifying manner using
the development into the homogeneous model. This is the motivation for the best
understanding of the notion of development in that general setting.
However, let us start with the classical definition of development into the tangent
bundle on a smooth manifold with affine connection. Let c be a smooth curve on M
with a fixed point x = c(to). Let X be the curve in the tangent space TXM which
arises by moving the velocity vectors c(to +1) back to TXM parallelly with respect
to the given affine connection. A choice of frame in TXM provides an identification
TXM = Rm
and we consider the development of c at x to be the integral (primitive)
curve of X in Rm
passing through the origin, i.e., it is the unique curve Y such that
Y' = X and Y(0) = 0. Obviously, this is a local construction and the development at
x = c(to) is defined only for such t where the parallel transport is available. This is
the classical definition of the development, as described in, e.g., [4, ch. Ill, § 5].
2000 Mathematics Subject Classification: 53B05, 53B15, 53C22.
Key words and phrases: Cartan geometry, development, generalized geodesies.
Supported by the grant 143100009 of the Czech Ministry of Education.
The paper is in final form and no version of it will be submitted elsewhere.
348 VOJTCCH 2ADNIK
Proposition 1.1 (well-known). The development of c at x = c(to) is a line segment
if and only if the vector field c(to +1) along c is parallel, i.e., the curve c is geodesic.
Following [5, 6, 7], we generalize the construction of development to all manifolds
endowed with a structure of Cartan geometry. At the same time, the notion of geodesies
generalizes as well so that the Proposition above is satisfied in that more general
setting. Development of curves gives rise to a correspondence between curves on the
base manifold of the Cartan geometry and curves in the homogeneous space. Under
some natural restriction we get a bijective correspondence which respects the order
of contact. Main aims of this work are to describe this correspondence precisely and
explicitly in the most general setting (Section 3) and to reformulate the general result
in the case of split Cartan geometries in order to distinguish generalized geodesies on
M in terms of their developments (Section 4). Theorem 3.2 is essential here, the rest
are consequences.
Acknowledgments. Thanks go to Jan Slovak for many inspirations and corrections,
and to the grant MSMT CR #MSM143100009 for the support. The final version of
this paper was compiled during author's short stay at the ESI institute in Vienna
in the occasion of the research program 'Geometric and analytic problems related to
Cartan connections'.
2. NOTATIONS AND PRELIMINARIES
Following [6], for a pair (G, H) of a Lie group G and a closed subgroup H C G, we
define Cartan geometry of type (G, H) on a smooth manifold M to be certain curved
analogy of the homogeneous bundle G -> G/H with the Maurer-Cartan form on G.
More precisely, denoting by g and f) the Lie algebras corresponding to given Lie groups,
Cartan geometry of type (G, H) on M consists of a principal H-bundle Q -> M with
a g-valued one-form u on Q which satisfies the following conditions:
(a) u(u) : TUQ -> g is a linear isomorphism for each u e Q)
(b) (rh
)*u = Ad(h~l
) ou for all he H,
(c) u(Cx(u)) = X for all X el).
Symbol Ad above means the adjoint representation of G on g and Cx is the fundamental
vector field of the principal right action of H on Q, generated by an element
Xe\).
The curvature of the Cartan geometry is defined as the 2-form K = du + |[(j, u] on
Q with values in g. Cartan geometry is called split if there is a fixed subalgebra n C g
complementary to \), i.e., g = n © f) on the level of vector spaces. Cartan geometry is
called reductive if there is a fixed complementary subspace n C g which is H-invariant,
i.e., invariant according to the adjoint representation Ad : G -> GL(Q) restricted to H.
For any X e g, the constant vector field u~l
(X) is the unique vector field on Q
which satisfies the condition u(u~l
(X)(u)) = X for all u e Q. Next, we can identify the
tangent bundle TM with the associated bundle Q XH (g/fj) where the representation of
the structure group H on g/f) is induced by the adjoint action on g. The identification
is provided by the prescription {u, X -f [)} h-> Tp-u~l
(X -f b)(u), where p denotes the
projection Q -> M.
In the case of split Cartan geometries, with the splitting g = n© fj, we identify g/f)
with n and via this identification the subalgebra n becomes an H-module. The action of
REMARKS ON DEVELOPMENT OF CURVES 3 4 9
the group H on n is called the truncated adjoint representation and it will be denoted
by Ad- Obviously, this action is determined by the condition Ad(h) = ~ ° Ad(ft) for
all h e H, where n is the projection g —> n in the direction of I). Now we can write
TM = Q xHn with respect to this action and the identification above turns into the
mapping {u,X} »-> Tp-uj~1
(X)(u) with the inverse f i-> {U,7T(OJ(£))}, where | G TUQ
is an arbitrary lift of the vector £ G TP(U)M. Further, any fixed subalgebra n gives
rise to a horizontal distribution o;_1
(n) on Q, hence each splitting enters a general
connection on Q. The equivariancy of u yields that this connection is principal if and
only if Ad = Ad, i.e., the geometry in question is reductive.
Example 2.1. As we suggested before, the most important example of Cartan geometry
of type (G, H) is the principal bundle G -> G/H together with the Maurer-Cartan
form on G. The structure equation says the curvature vanishes, so this is called flat,
or homogeneous, model of Cartan geometries of type (G,H). Much of the geometry
of the homogeneous model carries over to all Cartan geometries of the same type and
this is the main reason of its importance.
Example 2.2. Each manifold with an affine connection is a very obvious example of
Cartan geometry of type (G,H) where G = _4(m,R) is the group of affine motions
on Rm
and H = GL(m,R) is the stabilizer of origin. The bundle of linear frames
Pl
M plays the role of Q and the Cartan connection u G Q,l
(Pl
M, a(m,R)) is given
by 0 + 7, where 9 G fi1
(P1
M,Rm
) is the soldering form and 7 G nl
(Pl
M,Ql(m,R)) is
the principal connection corresponding to the linear connection on the tangent bundle.
This is an example of split and reductive Cartan geometry.
The homogeneous space G/H of affine geometry is Rm
and the left multiplication in
G restricted to H C GL(m, R) descends to the standard action on Rm
= G/H. Further,
the reductivity of the geometry means that the Ad-representation of G restricted to
H keeps the complement n = Rm
to \) invariant. It is easy to see this restricted action
coincides with the standard representation of H on Rm
= n, too.
3. CARTAN'S SPACE AND DEVELOPMENTS
For a Cartan geometry of type (G, H) with the principal bundle p : Q —> M, the
Cartaris space of M is the associated bundle SM = Q xH (G/H) with respect to the
action of H on G/H given by the left multiplication. The standard construction of
the so called extension Q of the principal bundle Q to the structure group G extends
uniquely the Cartan connection u G £ll
(Q, g) to the principal connection u G ttl
(Q}Q)
and this induces general connections on all bundles associated to Q with respect to the
action of H on the standard fiber given by the restriction of an action of the group G.
Hence we get a general connection on the Cartan's space SM = Q xH (G/H) =
Q xG(G/H) which will be essential for the construction of the development of curves.
More concretely, the extension of Q is the principal G-bundle Q = Q xH G, the
bundle Q is seen as a subbundle of Q due to the canonical reduction u .-•> {u, e}, and
the connection Q G fl^ff.g) is determined by the conditions to be principal and to
coincide with u on TQ c TQ. Altogether, the principal connection u satisfies:
(a) u(u)\Tug = UJ(U) : TUQ -> g for each ueQ,
(b) u(Cx(u)) = X for all X G g, u G Q,
350 VOJTECH 2ADNIK
(c) the horizontal lift of vector f = {u, X + f)} e TP(U)M is
t(u)=u-l
(X)(u)-(x(u)eTuQ.
First two conditions are obvious from definition. The third one is a consequence, in
particular, the lift | is well-defined, projects onto £, and £)(£) = X — X = 0. See, e.g.,
[7] for details.
Another ingredient for the construction of developments is the canonical global
section of the bundle projection SM -> M, given by the prescription x H> {U, o = eH}
for any u eQ staying over x e M. Indeed, the definition does not depend on a choice
of u E Q and the section is denoted by O. Hence, any fiber of the vertical tangent
bundle VSM = Q x# T(G/H) restricted to O(M) can be thought as an analogy of
the tangent space, since the bundles VSM\0(M) and TM are canonically isomorphic
via the isomorphism T0(G/H) = ajl). In this vein, the base manifold M seems to
be 'osculated' in each point by the homogeneous space G/H on the level of tangent
spaces.
Remarks. In the case of affine geometry, the Cartan's space SM coincides with the
tangent bundle TM since the standard fibers G/H and g/t) are the same Rm
and the
two actions of the structure group equal, as shown at the end of Example 2.2. Next,
the construction above reminds the osculation of non-degenerated real submanifolds
in C1
by quadrics which represent the modeling spaces of the induced CR-structure.
This notion goes really back to Cartan, see [5] for more details and comments.
Definition 3.1. For c being a parametrized curve on M.with a fixed point x = c(t0),
we define the development of c at x to be a curve devx c in the fiber SXM = G/H
as follows. First, the curve c can be seen as a parametrized curve, in the Cartan's
space SM due to the composition with the canonical section O. Then, locally, the
value (devxc)(t) is obtained by moving the point O(c(^o +1)) back to the fiber SXM
using the parallel transport induced by u. More precisely, if c(s) is the parallel curve
in SM determined by the initial condition c(0) = O(c(to +1)) and covering the path
s i-> c(tQ +1 + s) in M then (devx c)(t) = c(—t).
The whole construction works only on a neighbourhood I of t0 € R where the
parallel transport in SM along c is well-defined. Then, for J = I-t0y the development
devx c is a curve J -> SXM which maps 0 € J to 0(x) e SXM. For each u e p~x
(x),
the development of c at x can be represented by a unique curve c : J -> G/H mapping
0 £ J to the origin o e G/H so that (devx c)(t) = {u, c(t)}. Another frame uh e p~l
(x)
changes the curve to 4 - 1
° c. Fixing the frame u G P"1
(x), we often consider the
development at a; as a map which assigns to each curve c passing through x G M the
unique curve c in G/H as before.
Theorem 3.2. Let p : Q —> M be the principal bundle of Cartan geometry of type
(G,H). For a smooth curve c on M, fix a point x = c(t0) and arbitrary u € p~x
(x).
Let c be any lift of c in Q starting at u and let X be the curve in Q defined as
X(t) = w(jtc(t)). Then the development of c at x = p(u) is the curve (devxc)(£) =
{u, a(t)H} C SXM, where a is the unique curve in G satisfying
(1) 5a = K and o(0) = e.
REMARKS ON DEVELOPMENT OF CURVES 351
Symbol 5 means the left logarithmic, or Darboux, derivative determined by the
Maurer-Cartan form UJQ on G. In general, for a smooth map / : N -> G, the left
logarithmic derivative 5f = f*ujQ is a g-valued one-form on N, however, in the case of
dimN = 1, it is usually understood as a curve in g due to the identification with the
image of 1 6 TSR for all 5. Conversely, for a given one-form u : TN —•> g, one can ask
whether there is a smooth map / : N —> G such that u = 5f. The classical result of
Cartan shows that the only sufficient (and also necessary) condition for the existence
of local solutions is the structure equation du + \[u),u)] = 0 to be satisfied. Further,
if /i>/2 : N -» G are two maps such that 5f\ = 5f2 then there is a unique element
c G G such that /2 = 4 % Altogether, this is a non-abelian generalization of the
existence and uniqueness of the primitive function in elementary calculus—all proofs
and details can be found in [6, ch. 3, §§5-6].
Theorem 3.3 (Cartan). Let G be a Lie group with Lie algebra g and let g € G be a
fixed element. Let UJ be a g-valued one-form on the smooth manifold N satisfying the
structure equation duj+\[uj,uj) = 0. Then, for each point x E N, there is a neighborhood
U of x and a unique smooth map f :U -» G such that u\u = 5f and f(x) = g.
Next, for any smooth maps /1, f2 : N -» G, an analogy of the Leibniz rule for the
left logarithmic derivative has the form
(2) S(frf2)(x) = (5f2)(x) + Ad/2(l)-i oQfMx)
as shown in e.g. [3, p. 39]. Especially, for f2 = /f1
, the left hand side is the derivative of
the constant map to e G G, so we conclude that the following holds for any / : N —•> G
and all x e N,
(3) (5f-i)(x) = -Adf{x)o(5f)(x).
Proof of Theorem 3.2. We have to prove that the parallel curve over c in SM =
Q XG (G/H) starting at {u,a(t)H} attains the point 0(c(tQ +1)) = {c(t),eH} just
in the time t, where clearly c(to) = x = p(u). The essence of the proof is to describe
the parallel curve c in Q, covering c and starting at u = c(0) = c(0), with respect to
the principal connection Q as described at the very beginning of this section. Then
the condition above reads as {c(t),a(t)H} = {c(t),eH}, so we write.c(£) = cty-a-1
^)
for later use. Now we are going to determine a : J -» G so that c is the appropriate
parallel curve in Q, in particular, a(0) = e must be satisfied. *
The definition of X gives ftc(t) = u~l
(X(t))(c(t)) and the vector field £ta~l
(t)
is obviously written as T0£a-i(t)*(5a"l
)(t). Hence the derivative of c(t) = c(^)-o_1
(^)
yields
(4) |c(«) = Tra
-^-uj-l
{X(t))(c(t)) + Cffa-MW^W-o"1
©) •
The request for vectors ftc(t) to be horizontal .means that &(£tc(t)) = 0 for all t.
From the right equivariancy of the principal connection UJ we conclude that the latter
condition is equivalent to
(5) Ada{t)X(t) + (5a-1
)(t) = 0,
352 VOJTECH ZADNIK
since cD(Cx) = X for any X G g and u~l
(X(t))(c(t)) belongs to T^t)Q where u and u
coincide. According to (3), equation (5) just means
(6) X(t) = (5a)(t)
since the map Ad(a(t)) : g -» g is invertible for all t. Now, Theorem 3.3 guarantees
the (local) existence and uniqueness of a with the initial condition a(0) = e. This is
due to the fact that N is one-dimensional in our case, so all two-forms on N are trivial
and so the structure equation is trivially satisfied. •
Remarks, (a) Theorem 3.2 provides an alternative view on development of curves,
which can also serve as a definition, cf. [6, p. 209]. Although it is implicitly involved
in the statement, we now directly show, as an exercise, that the development of c at
x constructed in this way really does not depend on the choice of lift c. In the current
setting, let c2 = c-h be another lift of c starting at u G Q, i.e., h is a curve in H
such that h(0) = e. Then, considering the connection u on the principal H-bundle Q
instead of u on the G-bundle Q, the same arguments as in the proof above—namely,
the equivariancy of u applied to the derivative of c2(t) = c(t)-h(t)—shows that the
corresponding curve X2 in g is
(7) U{ici(t))=Adh-Ht)X{t) + (6h)(t)
where X = 5a. The resulting development is then represented by the unique curve a2
in G such that 5a2 = X2 and 02(0) = e. But, 5(a-h) = X2 and a(0)-h(0) = e, hence
a2 = a-h by the uniqueness of the solution. Curves a and a2 share the same projection
to G/H, so the development is well-defined in this way.
(b) Next, Theorem 3.2 clearly demonstrates the property which everybody expects
developments in the homogeneous model to have, namely, that developments of curves
in M = G/H into the fiber of the Cartan's space SM = G xH (G/H) are just the
initial curves but shifted to the origin. Indeed, for a smooth curve c(t) = b(t)H in
M = G/H and a point x = c(t0)> let c(t) = b(t 4- t0) be the lift of c in G starting
at u = b(t0). By the definition of X, we get X = 5c and, according to u G G, the
development of c is represented by the unique solution of 5a = X with the initial data
a(0) = e. But, c = £uoa since 6(0) =u = b(t0), hence the resulting curve in G/H is a
shift of c according to
(8) a(t)=b(t0)-l
-b(t + t0).
Another representative b2 of c leads to the same curve in G/H but conjugated by
h = b(t0)~l
b2(t0) G H, as in the general case. In particular, for curves which map 0
to the origin o = eH, the developments at 0 with respect to e G p_1
(o) are absolutely
unchanged.
Back to the general picture, the construction of developments establishes a correspondence
between smooth curves on M which map 0 to a fixed point x G M and
smooth curves in the fiber SXM of the Cartan's space mapping 0 to 0(x). Now we are
going to show this correspondence is bijective, Proposition 3.5.
First of all, we formulate a slight modification of Theorem 3.3 where we consider a
general Cartan geometry with the total space Q and Cartan connection u G ^(QiQ),
instead of the Lie group G with the Maurer-Cartan form UQ G fl^G-g), and the
pullback f*u, instead of the left logarithmic derivative 5f = f*UG- We further consider
REMARKS ON DEVELOPMENT OF CURVES 353
dim N = 1, which is also the only possibility allowing an analogy to Theorem 3.3 in the
case of general Cartan geometry with non-vanishing curvature. In that case, fl-valued
one-forms are seen as curves in g, the structure equation is trivially satisfied, hence we
get the following formulation.
Lemma 3.4. Let G -> M be the principal bundle of the Cartan geometry with the
Cartan connection u e ttl
(G,Q) and let u e G be afixedframe. Let X be a smooth
curve in g. Then, for each to e R, there is a neighborhood I of to and a unique smooth
curve c: I -> G such that X\j = u(c) and c(to) = u.
Proof. In order to prove this statement, it suffices to follow the proof of Theorem
3.3 as written in, e.g., [6, pp. 116-118]. The essence of that proof is a construction of
the graph of / in N x G. Locally, this graph is found as the leaf through the point
(x, g) of some integrable distribution of rank = dim N on N x G. The distribution
is determined by the one-form u on N and the Maurer-Cartan form WQ, and the
integrability is guaranteed by the fact that the structure equation is satisfied both for
UQ and u by the assumption.
In the curved case, all constructions match as well but the integrability of the distribution
in question is no longer satisfied in general, except the case with dim N = 1. •
Now we see that Lemma 3.4 provides the inverse to the map assigning to any smooth
curve c on M with c(0) = x the representative of its development devx c with respect
to a fixed frame u e p~1
(x). The restriction to curves mapping 0 to a; is clear since,
for any reparametrization of the form <p(t) = t + tQ} the curves c and cotp have got
the same development. Altogether, we obtain the following identification.
Proposition 3.5. For afixedu eG, the development of curves on M into the homogeneous
space G/H establishes a bijective correspondence between the sets
{curves on M mapping 0 to x = p(u)} *--- {curves on G/H mapping 0 to o = eH} .
Proof. The right arrow is the development with respect to u e G as described above.
We have only to describe the left arrow in more details so that it is the inverse to the
map before.
Let c be a curve in G/H which maps 0 to the origin, and let o : I -> G be any lift
of c such that o(0) = e. For thefixedueG, there is a unique curve Ci in G such that
ci(0) = u and u;(ci) = 5a, due to Lemma 3.4. Thus, we get the projection p o ci to M
as the image of c. Of course, the result does not depend on the choice of a, provided
that a(0) = e. The rest is clear, namely, we have really constructed the inverse to the
map in question. •
The identification above is not canonical at all but, by the definition of development,
it depends onfixedueG over x so that another frame uh e p~l
(x) changes the image
of the right arrow by the conjugation of h E H. As an exercise, one can identify the
set of curves on M mapping 0 to afixedpoint x e M with the set of conjugacy classes
of curves on G/H mapping 0 to the origin. Anyway, the bijection above is compatible
with taking jets in 0 as follows.
Proposition 3.6. Let Ci and c2 be smooth curves in M such that Ci(0) = x = c2(0).
Then, for any positive integer r, jjci = j$c2 if and only ifJo(devx ci) = Jo(devx c2).
Proof. Fix a frame ueG over x = Ci(0) = c^O). On a neighborhood U of x e M,
let us consider a local section a : U -> G mapping x to u. For i = 1,2 and small £,
3 5 4 VOJTfiCH 2ADNIK
the curve c» = a o a is a lift of c* with Cj(0) = u. Since a is a diffeomorphism onto the
image, the following equivalence is clear: jjci = JQC2 if and only if jjci = JQC2.
Let Xi(t) = u(jtCi(t)) be the curves in g, as in Theorem 3.2. The form u :TQ -» g
is an absolute parallelism, and £-jet of Xi is expressed in terms of £-jet of CJ and
(^ + l)-jet of Cj, hence, it is obvious that if jjci = JoC2 then J5_1
Ki = Jo"1
^-- For
the converse implication, let as assume the equality JQ~1
X\ = JQ~1
X2 to be satisfied.
Clearly, if Xi(0) = X2(0) then JQCI = JQC2 and, inductively, the equality JQC\ = JQC2
implies Jo+1
^i = Jo*1
^ for alH = 1,..., r - 1. Altogether: fidi = jjo. -f and only if
Jor
"%=Jo"%.
Next, let oi and a2 be the unique curves in G such that Sai = Xi and Oj(0) = e.
By the definition of the left logarithmic derivative, the condition Sai = X\ reads
as WG(^OJW) = Xi(t), for the Maurer-Cartan form uG : TG -» 9, and the same
arguments as in the latter paragraph yield: JQ~1
XI = JQ~1
X2 if and only if jjoi = Joa
2Finally,
r-th jet prolongation of the projection G -» G/H ensures the implication:
if Jo°i = 3oa
2 then ,7o(devxci) = Jo(devx c2). Altogether, we get:
JQCI = JQC2 = > jj(devz a) = J5(devx c2).
The converse implication can be proved just in the same way as before following
the inverse to the construction of development. Starting with a local section a of the
projection G -> G/H such that 0(0) = e, we consider a* to be the lift of devzCi in
the image of a, and so forth. The essential point here is the bijective correspondence,
established by Lemma 3.4, between curves o in G mapping 0 to e and curves c in Q
mapping 0 to fixed u G Q such that Sa = v(£tc). In previous paragraphs, we have
proved via Xi = Sai = u
(diCi) that jJa
i = Joa
2 if and only if JQC\ = JQC2. The rest is
obvious. •
4. SPLIT GEOMETRIES AND GENERALIZED GEODESICS
Following Theorem 3.2, the development of c at x = p(u) is represented by a curve
a in G which is the solution of Sa = X with o(0) = e. In the case of split Cartan
geometries, the image of the complement n C g under the map exp : g-> G forms a
subgroup in G complementary to fibers of the projection G -> G/H, locally around
the origin. In that case, the development above can be described by a unique curve Y
in n so that a(t) = exp(y(J)) mod H, in other words, this is just expression of a in
the normal coordinates through e € G. Further, if we consider the horizontal lift of c
instead of a general lift c then the image of X belongs to n and the entire problem of
developments can be solved in n as follows.
Considering the curve o of the form o = exp oY, we have to describe <J(exp oY)
which is (S exp) o TY by the definition of 5. So we are going to find the formula
for tfexp : T& -» g first. For each linear endomorphism z 6 gl(g) let us denote by
g(z) = ----=--- the formal power series YlkLo Tk+Tjl**- ^ e n
^ e
formu
la
f°r
the right
logarithmic derivative deduced in [3, 4.27] can be easily adapted for the left one so
that equality
(9) (*exp)(y) = g(*d(-Y)) :TYQ->Z
holds for any Y G Q. In particular, for Y : I -> 0, the left logarithmic derivative
of the composition exp oY : I -.> G, understood as a curve in g, equals to
REMARKS ON DEVELOPMENT OF CURVES 3 5 5
g(a.d(-Y(t)))(Y'(t)). Hence, omitting the variable t, equation (1) reads as
(10
) £7rb!(ad(
~y))
*(y
')=
*»
A,=0 * '
which is afirst-orderODE with the initial condition Y(0) = 0. Considering horizontal
lift of c, i.e., considering X C n, the latter equation is then put in n.
Remarks. Especially, if the subalgebra n C g is abelian then the equation above takes
the very nice and easy form
(11) Y' = X.
This happens, for example, in the case of irreducible parabolic geometries whilst, in
general parabolic geometries, the complement n C gis only nilpotent. Anyway, the
sum in (10) is always finite in these cases.
Another instance of a geometry with n abelian is the affine geometry where the
formula above just recovers the definition of development at the beginning of Section 1.
Indeed, under the identification TM = Q XH n with the notation above, the velocity
vector field along c is written as c(t) = {c(t),X(t)}} and the definition of parallel
displacement in associated bundles provides the equivalence of these two concepts. In
the sequel, we conclude the analogy with the affine case by Proposition 4.2 which is
a straight generalization of that in 1.1. Following [6, p. 210], we first generalize the
notion of geodesies to any split Cartan geometry.
Definition 4.1. Let p : Q -» M be a Cartan geometry of type (G,-?), split as g =
n © f), and let A C n be an arbitrary subset. Smooth curve on the base manifold is
called a generalized geodesic of type CA if it has the shape
c^(f)=P(Fir1(x)
(u))
for some u€Q and X G A.
Indeed, affine geodesies can be described just in the same way in view of Example
2.2, cf. [4, p. 139]. Further, this definition recovers some well-known types of curves in
particular geometries, e.g., the conformal circles in conformal geometries, the ChernMoser
chains in hypersurface CR-geometries, and others.
The generalized geodesic cu
'x
clearly goes through the point p(u) = c(0) with the
tangent vector {u,X} e TP(U)M but, in contrast to the affine (more generally, reductive)
case, this initial condition does not determine the generalized geodesic uniquely.
In other words, another representative of the tangent vector may define another curve,
so we need initial conditions of higher order in order the generalized geodesies to be
uniquely specified. Furthermore, according to the choice of A C n, there appear curves
of various types on the base manifold, which may behave rather different. The very
well-known instance of such curves are null-geodesies and null-chains on conformal and
CR-manifolds of indefinite signature, respectively.
The notion of development is often used to distinguish curves on all manifolds endowed
with a Cartan geometry of fixed type by means of distinguished curves in the
homogeneous model. The following Proposition provides an alternative definition of
generalized geodesies.
356 VOJTECH 2ÁDNÍK
Proposition 4.2. The development of c at x = c(tQ) has the form (dev^c)^) =
{u}exp(tX)H} C SXM, for some X G A and u € Q, if and only if c is a geodesic of
type CA.
Proof. According to Theorem 3.2 and conventions from the beginning of this section,
we have only to prove that the curve X(t) is constant if and only if Y(t) = tX for
some X 6 A, where Y is the solution of (10) with the initial condition Y(0) = 0.
If X(t) = X is the constant curve, we define Y(t) = tX and, conversely, if Y has the
form Y(t) = tX then we put X(t) = X. Anyway, equation (10) is still satisfied, since
the sum YlkLi (*+!)! (ac
^("'^))fc
(X) vanishes for all t and so the only nonzero term on
the left hand side is X. The uniqueness of the development completes the proof. •
Remarks. All the observations above are essential for closer study of some basic
properties of generalized geodesies, which can be read now on the level of the homogeneous
space using their developments. This is presented in [2] in the case of parabolic
geometries.
Concerning parabolic geometries, the notion of generalized Weyl structures [1] leads
to a class of affine connections on the base manifold, which underlie the given Cartan
connection in some sense. Then besides the development of curves into the homogeneous
space of the parabolic geometry, we have got the classical development into the
tangent space with respect to a fixed affine connection from the class in question. In
this vein, it is natural to look for the relationship between these two developments considering
a fixed Weyl structure. For conformal and projective geometries, this problem
is fully solved by Tanaka in [8, 9] in terms of the so called rho-tensor. Following the
process of Theorem 3.2 above, one can easily generalize these results to an arbitrary
parabolic geometry. We wish to come back to this topic elsewhere.
REFERENCES
[1] A. Cap, J. Slovdk, Weyl structures for parabolic geometries, Math. Scand. 93 (2003), no. 1,53-90.
[2] A. Cap, J. Slovak, V. Z&dnfk, On distinguished curves in parabolic geometries, Transform.
Groups 9 (2004), no. 2,143-166.
[3] I. Kolaf, P. W. Michor, J. Slovdk, Natural operations in differential geometry, Springer-Verlag,
1993.
[4] S. Kobayashi, K. Nomizu, Foundations of differential geometry I, Interscience Publishers, 1963.
[5] L. K. Koch, Development and distinguished curves, preprint (1993).
[6] R. W. Sharpe, Differential geometry: Cartan's generalization of Klein's Erlangen program, Grad
uate Texts in Mathematics 166, Springer-Verlag, 1997.
[7] J. Sloviik, Parabolic geometries, research lecture notes, IGA preprint 97/11,1997.
[8] N. Tanaka, Conformal connections and conformal transformations, Trans. Amer. Math. Soc. 92
(1959), 168-190.
[9] N. Tanaka, Projective connections and projective transformations, Nagoya Math. J. 12 (1957),
1-24.
DEPARTMENT OF ALGEBRA AND GEOMETRY, MASARYK UNIVERSITY
JANÁČKOVO NÁM. 2A, 662 95 BRNO, CZECH REPUBLIC
E-mail: zadnikOmath.muni. cz