Geodesic Reduction via Frame Bundle Geometry
A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant...
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| description | A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 020, 17 pages
Geodesic Reduction via Frame Bundle Geometry
Ajit BHAND
Department of Mathematics, University of Oklahoma, Norman, OK, USA
E-mail: abhand@math.ou.edu
Received October 12, 2009, in final form February 18, 2010; Published online February 22, 2010
doi:10.3842/SIGMA.2010.020
Abstract. A manifold with an arbitrary affine connection is considered and the geodesic
spray associated with the connection is studied in the presence of a Lie group action. In
particular, results are obtained that provide insight into the structure of the reduced dy-
namics associated with the given invariant affine connection. The geometry of the frame
bundle of the given manifold is used to provide an intrinsic description of the geodesic spray.
A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift
of the symmetric product is obtained, which provides a key to understanding reduction in
this formulation.
Key words: affine connection; geodesic spray; reduction; linear frame bundle
2010 Mathematics Subject Classification: 53B05; 53C05; 53C22; 58D19
1 Introduction
The geometry of systems with symmetry has been an active area of research in the last several
years. The study of manifolds with certain special geometric structure invariant under a Lie
group action leads to what is known as reduction theory. Such questions arise in, for example,
geometric mechanics. In this framework, the presence of symmetry allows the dynamics on
a manifold to be studied on a lower dimensional manifold. In mechanics, there are at least
three different ways of describing dynamics on a manifold, corresponding to the Lagrangian,
Hamiltonian and affine connection formulations respectively. While the reduction theory for
Lagrangian and Hamiltonian systems has been well developed (see [1, 9, 10, 20]), these results
have been obtained by using variational analysis and symplectic geometry respectively. The
main reason behind following this approach is the fact that the dynamics for such systems arises
from variational principles which are manifested by symplectic structures in the Hamiltonian
framework. However, when the dynamics on a manifold are given in terms of the geodesic
equation of an affine connection, we cannot use variational analysis unless additional structure,
such as a metric, is provided. Mechanical systems for which the dynamics are given by the
geodesics of an affine connection that is not Levi-Civita include systems subjected to velocity
constraints (see, for example, [17, 18] and Section 5.4).
We consider an arbitrary affine connection on a manifold invariant under the action of a Lie
group and provide results that enable us to decompose the reduced geodesic spray corresponding
to the affine connection using tools from differential geometry only. In other words, we do not
use variational methods. In arriving at our results, we come to a deeper understanding of the
geometry of bundle of linear frames and its relationship with the geometry of the tangent bundle
of the given manifold.
The setup we consider is the following. Let M be a manifold and G a Lie group which acts
on M in such a manner that M is the total space of a principal bundle over M/G. The Lie
group G also acts on the bundle L(M) of linear frames over M via the lifted action. It is known
that there is a one-to-one correspondence between principal connections on L(M) and affine
mailto:abhand@math.ou.edu
http://dx.doi.org/10.3842/SIGMA.2010.020
2 A. Bhand
connections on M [15]. Let ω be a G-invariant principal connection on L(M) with ∇ the corre-
sponding affine connection on M . The geodesic spray Z corresponding to ∇ is a second-order
vector field on the tangent bundle TM with the property that the projection of its integral
curves correspond to geodesics on M . Thus, to understand how the geodesics evolves under
symmetry, Z is the appropriate object to study. Since additional structure is not available, we
exploit the geometry of the linear frame bundle in order to fully understand the meaning of the
geodesic spray (which is classically defined in local coordinates). The first significant step in
this direction is to provide an intrinsic definition of the geodesic spray that uses frame bundle
geometry. We are able to provide such a definition.
Moving ahead, we give a new interpretation of the geodesic invariance of a distribution on
the manifold M using frame bundle geometry and provide a new proof of a characterization due
to Lewis [17] using the symmetric product.
Next, we turn our attention to understanding the reduced geodesic spray of a given con-
nection. Our main idea is that it is possible to study reduction using only geometric data
and without variational analysis. To our knowledge, the proposed approach of using frame
bundle geometry to study reduction is new. Even though the reduction method is discussed in
the context of the geodesic spray, it can be applied to a general invariant second-order vector
field.
The reduction procedure presented in this paper is based on the author’s thesis [3]. Bullo and
Lewis did some preliminary work in this direction [6] and recently, Crampin and Mestdag [13]
have presented an approach which is similar in spirit to ours. They consider reduction and
reconstruction of general second-order systems and provide a decomposition of the reduced
system into three parts.
The definition of the geodesic spray using frame bundle geometry, provided in Section 3,
enables us to find a formula relating the geodesic spray and the vertical and complete lifts.
In Section 4 we present our reduction methodology along with the main result (Theorem 4.8)
which provides a new coordinate-free way of decomposing the reduced geodesic spray in terms of
objects defined on the reduced space. Finally, in Section 5 we provide a geometric interpretation
of the result in the Riemannian case and discuss some avenues for future work.
2 Linear connections
In this section we review some concepts relevant to our investigation and establish notation to be
used throughout the paper. Let M be an n-dimensional smooth manifold and L(M)(M,GL(n;R))
the bundle of linear frames with total space L(M), base space M , structure group GL(n; R) and
canonical projection πM . We denote the (right) action of GL(n; R) by Φ : L(M)×GL(n; R) →
L(M). For fixed a ∈ GL(n; R), this action induces a map Φa : L(M) → L(M) given by
Φa(u) = Φ(u, a). Recall that a linear frame u at x ∈ M is an ordered basis (X1, X2, . . . , Xn) of
the tangent space TxM . If a = (ai
j) ∈ GL(n; R) and u = (X1, X2, . . . , Xn), then ua := Φa(u) is
the linear frame (Y1, Y2, . . . , Yn) at x defined by Yi =
∑n
j=1 aj
iXj . Equivalently, if (e1, e2, . . . , en)
is a standard basis for Rn, a frame u = (X1, X2, . . . , Xn) at x can also be defined as a linear
isomorphism u : Rn → TxM given by uei = Xi. In other words, if ξ =
∑n
i=1 ξiei ∈ Rn,
then uξ =
∑n
i=1 ξiXi. This is the notion of a linear frame that we will use throughout the
paper. The infinitesimal generator corresponding to an element A ∈ gl(n, R) of the Lie algebra
will be denoted by AL(M). The canonical form θ of L(M) is a one-form on L(M) defined by
θ(Xu) = u−1 (πM (Xu)) , Xu ∈ TuL(M).
A principal connection ω in the bundle L(M)(M,GL(n; R)) of linear frames over M is called
a linear connection on M . The associated horizontal subbundle is denoted by HL(M). The tan-
gent bundle of M , denoted by τM :TM→M , is a bundle associated with L(M)(M,GL(n;R)) [15].
Geodesic Reduction via Frame Bundle Geometry 3
Given a linear connection ω on M , for each ξ ∈ Rn, the standard horizontal vector field
corresponding to ξ, denoted by B(ξ) : L(M) → TL(M), is defined as follows. For each u ∈ L(M),
the vector B(ξ)u is the unique horizontal vector at u with the property that TuπM (B(ξ)u) = uξ.
We consider an arbitrary principal bundle P (M,G) with total space P , base space M and
structure group G. The canonical vertical bundle will be denoted by V P .
Given a principal fiber bundle P (M,G) and a representation ρ of G on a finite-dimensional
vector space V , a pseudotensorial r-form of type (ρ, V ) on P is a V -valued r-form ϕ on P such
that
Φ∗
gϕ = ρ
(
g−1
)
· ϕ, g ∈ G,
where Φ is the action of G on P . A pseudotensorial r-form ϕ of type (ρ, V ) is called a tensorial
r-form if it is horizontal in the sense that ϕ(X1, . . . , Xr) = 0 whenever Xi is vertical for at least
one i ∈ {1, . . . , r}. A connection one-form ω on a principal bundle P (M,G) is a pseudotensorial
one-form of type (Ad(G), g), where Ad(G) is the adjoint representation of G on g. The following
result characterizing the set of all principal connections on P can be readily proved.
Proposition 2.1. Let ω be a principal connection one-form on a principal bundle P (M,G) and
let α be a tensorial one-form of type (Ad(G), g) on P . Then ω̄ := ω + α defines a new principal
connection on P . Conversely, given any two principal connection forms ω and ω̄ respectively,
the object α := ω̄ − ω is a tensorial one-form of type (Ad(G), g) on P .
In other words, the space of principal connections is an affine space modeled on the vector
space of tensorial one-forms of type (Ad(G), g).
To each vector field Y on M we can associate a function fY : L(M) → R as follows. For
u ∈ L(M), we have fY (u) = u−1(Y (πM (u))). The following result provides a correspondence
between tensorial one-forms on L(M) and (1, 2) tensor fields on M .
Proposition 2.2. There is a one-to-one correspondence between tensorial one-forms of type
(Ad(GL(n; R)), gl(n; R)) on L(M) and (1, 2) tensor fields on M .
Proof. Since the tangent bundle τM : TM → M is the bundle associated with L(M) with
standard fiber Rn, for each (1, 2) tensor field S on M , and u ∈ L(M), we can define a map
αS : TL(M) → gl(n; R) as follows. Let X̃ ∈ TuL(M), for u ∈ L(M) and let η ∈ Rn. Then
u
(
αS(u)(X̃)η
)
= S
(
TuπM (X̃), uη
)
. (2.1)
Since αS(X̃) ∈ gl(n; R), the product α(X̃)η ∈ Rn. We now show that αS is a tensorial form of
type (Ad(GL(n; R)), gl(n; R)). For a ∈ GL(n; R), let Ỹ = TuΦaX̃ ∈ TuaL(M). Then, using the
definition (2.1), we get
(ua)
(
αS(ua)(Ỹ )η
)
= S
(
TuaπM (Ỹ ), uaη
)
= u
(
αS(u)(X̃)(aη)
)
from which we get
αS(ua)(Ỹ ) = a−1αS(u)(X̃)a,
which means that αS is pseudotensorial. Next, if X̃ ∈ VuL(M), it is easy to see that αS(X̃) = 0,
which shows that αS is tensorial.
Conversely, given a tensorial one-form α : TP → gl(n; R), we can define a (1, 2) tensor
field Sα as follows:
Sα(X, Y ) = u
(
α(X̃)fY (u)
)
, X, Y ∈ TxM, πM (u) = x,
where X̃u ∈ TuL(M) has the property that TuπM (X̃) = X. Since α is tensorial, Sα is well-
defined. �
4 A. Bhand
Corollary 2.3. Let ω and ω̄ be linear connections of M and let ∇ and ∇, respectively, be the
corresponding covariant derivatives. If α = ω̄ − ω then, for vector fields X and Y on M , we
have
∇XY = ∇XY + Sα(X, Y ),
where Sα is the unique (1, 2) tensor field on M corresponding to α.
This result, therefore, characterizes the set of all affine connections on the manifold M .
3 The geodesic spray of an affine connection
In this section we study the geodesic spray associated with a given affine connection. This object
is typically defined in terms of local coordinates and here we provide an intrinsic definition using
the geometry of the linear frame bundle. Given a linear connection ω on M , for fixed ξ ∈ Rn,
let Φξ : L(M) → TM be the association map given by Φξ(u) = uξ. We define a (second-order)
vector field Z : TM → TTM called the geodesic spray as follows:
Z(v) = TuΦξ(B(ξ)u), v ∈ TM, (3.1)
where u ∈ LτM (v)(M) and ξ ∈ Rn are such that uξ = v, and B(ξ) is the standard horizontal
vector field corresponding to ξ for the linear connection ω associated with ∇. We have the
following result.
Proposition 3.1. The map Z defined in (3.1) is a second-order vector field on TM . The
coordinate expression for Z, in terms of the canonical tangent bundle coordinates (xi, vi) is
given by
Z = vi ∂
∂xi
− Γi
jkv
jvk ∂
∂vi
. (3.2)
Proof. We first show that Z as given by (3.1) is well-defined. The canonical projection on
the tangent bundle is denoted by τM : TM → M . For a given v ∈ TM , we write x := τM (v).
Suppose that u′ ∈ Lx(M) and ξ′ ∈ Rn are such that u′ξ′ = v = uξ. Then, we must have u′ = ua
for some a ∈ GL(n; R). Consequently, ξ′ = a−1ξ. We compute
TuaΦa−1ξ
(
B(a−1ξ)ua
)
= TuaΦa−1ξTuRa(B(ξ)u) = Tu(Φa−1ξ ◦Ra)B(ξ)u = TuΦξ(B(ξ)u),
where the first equality follows from the properties of a standard horizontal vector field. Let us
now show that Z is a second-order vector field. We have
TτM (Z(v)) = TτM (TuΦξB(ξ)u) = Tu(τM ◦ Φξ)B(ξ)u = Tu(πM )(B(ξ)u) = uξ = v
as desired.
It now remains to be shown that the coordinate representation of Z is as given in (3.2), but
this follow directly from the coordinate representation of B(ξ). �
3.1 Tangent and vertical lifts
If X is a vector field on M we can define a unique vector field X̃ on L(M) corresponding to X
as follows. Let φX
t be the flow of X. The tangent lift XT is a vector field on TM defined by
XT (vx) =
d
dt
∣∣∣∣
t=0
TφX
t (vx).
Geodesic Reduction via Frame Bundle Geometry 5
Let u ∈ Lx(M) and ξ ∈ Rn be such that uξ = vx. For ξ fixed, let Φξ : L(M) → TM be the
association map. The flow of XT defines a curve ut in L(M) by ut = TxφX
t · u. That is,
Φξut =
(
TxφX
t ◦ Φξ
)
u.
The map Φ̃t(u) = ut defines a flow on L(M). The corresponding vector field is called the natural
lift X̃ of X onto L(M). Thus, we have
XT (vx) = TuΦξX̃(u).
Given vx, wx ∈ TxM , the vertical lift of w at v is defined by
vlftvx(wx) =
d
dt
∣∣∣∣
t=0
(vx + twx).
3.2 Decomposition of the geodesic spray
The following result provides an explicit relationship between ∇ and the connection one-form ω
of the corresponding linear connection.
Proposition 3.2. Let M be a manifold with a connection ∇ with the corresponding linear
connection one-form ω. Given vector fields X and Y on M , let X̃ be the natural lift of X onto
L(M), and fY : L(M) → Rn the function associated with Y . Then,
∇XY (x) = [X, Y ](x) + u
(
ω(X̃(u))fY (u)
)
, πM (u) = x. (3.3)
Proof. Let us first verify that the right-hand side of (3.3) is independent of the choice of
u ∈ Lx(M). For a ∈ GL(n; R), we compute
(ua)
(
ω(X̃ua)fY (ua)
)
= (ua)
(
ω(TuΦaX̃u)a−1fY (u)
)
= (ua)
(
a−1ω(X̃u)a(a−1fY (u))
)
= u
(
ω(X̃u)fY (u)
)
.
We shall now prove that
LXhfY (u) = u−1 ([X, Y ](x)) + ω
(
X̃(u)
)
fY (u).
Notice that hor(X̃) = Xh since both are horizontal vector fields on L(M) projecting to X. In
other words,
Xh = X̃ − (ω(X̃))L(M),
where ω(X̃)L(M) is the vertical vector field on L(M) given by
u 7→ ω(X̃(u))L(M)(u).
Therefore,
LXhfY (u) = L
X̃
fY (u)−L
ω(X̃)L(M)
fY (u). (3.4)
The flow of X̃ is ΦX̃
t (u) = TxΦX
t · u, where ΦX
t is the flow of X and x = πM (u). The first term
on the right-hand side of (3.4) is
L
X̃
fY (u) =
d
dt
fY
(
TxΦX
t · u
)∣∣∣∣
t=0
=
d
dt
(
u−1TΦX
t (x)Φ
X
−t
)(
Y (ΦX
t (x))
)∣∣∣∣
t=0
=
d
dt
u−1Φx
t
∗Y (x)
∣∣∣∣
t=0
= u−1[X, Y ](x).
6 A. Bhand
We next compute the second term on the right-hand side of (3.4)
L
ω(X̃)L(M)
fY (u) =
d
dt
fY
(
u exp
(
tω(X̃(u))
))∣∣∣∣
t=0
=
d
dt
(
exp
(
− tω(X̃(u))
)
u−1Y (x)
)∣∣∣∣
t=0
= −ω(X̃(u))u−1Y (x) = −ω(X̃(u))fY (u),
where the second equality above follows from the fact that fX is a pseudotensorial form of degree
zero. This completes the proof. �
Remark 3.3. It is known that every derivation D of the tensor algebra of M can be decom-
posed as
D = LX + S,
where X is a vector field on M and S is a (1, 1) tensor field on M . From the previous result, it
follows that the (1, 1) tensor field associated with the derivation ∇X is given by
TxM 3 v 7→ u
(
ω(X̃(u))fYv(u)
)
, πM (u) = x,
where Yv is any vector field on M with value v at x.
We now prove the main result of this section.
Proposition 3.4. Let v ∈ TxM for some x ∈ M , and Xv be an arbitrary vector field that has
the value v at x. Then,
Z(v) = (Xv)T (v)− vlftv(∇XvXv(x)). (3.5)
Proof. Using local coordinates xi around x in M , we write Xv = Xv
i ∂
∂xi . Then,
(Xv)T (v) = vi ∂
∂xi
+ vj ∂Xv
i
∂xj
∂
∂vi
,
and
∇XvXv(x) =
(
∂Xv
i
∂xj
Xv
j + Γi
jkXv
jXv
k
)
∂
∂xi
=
(
∂Xv
i
∂xj
vj + Γi
jkv
jvk
)
∂
∂xi
.
So
vlftv(∇XvXv(x)) =
(
∂Xv
i
∂xj
vj + Γi
jkv
jvk
)
∂
∂vi
.
Thus,
(Xv)T (v)− vlftv(∇XvXv(x)) = vi ∂
∂xi
+ vj ∂Xv
i
∂xj
∂
∂vi
−
(
∂Xv
i
∂xj
vj + Γi
jkv
jvk
)
∂
∂vi
= vi ∂
∂xi
− Γi
jkv
jvk ∂
∂vi
= Z(v).
This proves the result. Notice that, even though each of the two terms XT
v (v) and vlftv(∇XvXv)
depends on the extension Xv, the terms that depend on the derivative of Xv cancel in the
expression for Z.
Geodesic Reduction via Frame Bundle Geometry 7
Alternate proof. Given vx ∈ TxM , let u ∈ LxM and ξ ∈ Rn be such that uξ = vx. Let Xv be
a vector field with value vx at x and fXv : L(M) → Rn be the corresponding function on L(M).
Then, for each u ∈ L(M),
hor(X̃v(u)) = B(fXv(u))u,
where X̃v is the natural lift of Xv onto L(M). We have
Z(Xv(x)) = TuΦfXv (u)B(fXv(u))u = TuΦfXv (u)
(
X̃(u)− ω(X̃(u))L(M)
)
= XT
v (Xv(x))− TuΦfXv (u)
(
ω(X̃(u))L(M)
)
= XT
v (Xv(x))− d
dt
ΦfXv (u)u exp
(
tω(X̃(u))
)∣∣∣∣
t=0
= XT
v (Xv(x))− vlftXv(x)u
(
ω(X̃(u))fXv(u)
)
= XT
v (Xv(x))− vlftXv(x)∇XvXv(x),
where we have used Proposition 3.2 in the last step. �
3.3 Geodesic invariance
We recall the notion of geodesic invariance.
Definition 3.5. A distribution D on a manifold M with an affine connection ∇ is called
geodesically invariant if for every geodesic c : [a, b] → M , ċ(a) ∈ Dc(a) implies that ċ(t) ∈ Dc(t)
for all t ∈ [a, b].
It turns out that geodesic invariance can be characterized by studying a certain product on
the set of vector fields on M . Let M be a manifold with a connection ∇. Given vector fields
X, Y ∈ Γ(TM), the symmetric product 〈X : Y 〉 is the vector field defined by
〈X : Y 〉 = ∇XY +∇Y X. (3.6)
The following result gives a description of the symmetric product using linear frame bundle
geometry.
Proposition 3.6. Let M be a manifold with a connection ∇, let ω be the associated linear
connection and θ the canonical form on L(M) respectively. If X and Y are vector fields on M
and X̃ and Ỹ the respective natural lifts on L(M), then
〈X : Y 〉(x) = 2u
(
Sym(ω ⊗ θ)(X̃, Ỹ )
)
, πM (u) = x. (3.7)
Proof. It is clear that the right-hand side of (3.7) is independent of the choice of u ∈ Lx(M).
By definition, we have θ(X̃(u)) = fX(u) and θ(Ỹ (u)) = fY (u). We compute
2u
(
Sym(ω ⊗ θ)(X̃(u), Ỹ (u))
)
= u
(
ω(X̃(u))θ(Ỹ (u)) + ω(Ỹ (u))θ(X̃(u))
)
= u
(
ω(X̃(u))fY (u) + ω(Ỹ (u))fX(u)
)
.
The result now follows from Proposition 3.2. �
Remark 3.7. The object sym(ω ⊗ θ) defines a quadratic form Σu : TuL(M) × TuL(M) →
TuL(M), u ∈ L(M) as follows:
Σu(X, Y ) = Z̃X,Y (u), X, Y ∈ TuL(M),
where Z̃X,Y is the natural lift onto L(M) of the vector field ZX,Y on M given by
ZX,Y (x) = u(sym(ω ⊗ θ)(X, Y )), π(u) = x.
This is seen to be well-defined.
8 A. Bhand
Given a distribution D on M , we represent by Γ(D) the set of vector fields taking values in D.
The following result, proved by Lewis [17], provides infinitesimal tests for geodesic invariance
and gives the geometric meaning of the symmetric product.
Theorem 3.8 (Lewis). Let D be a distribution on a manifold M with a connection ∇. The
following are equivalent:
(i) D is geodesically invariant;
(ii) 〈X : Y 〉 ∈ Γ(D) for every X, Y ∈ Γ(D);
(iii) ∇XX ∈ Γ(D) for every X ∈ Γ(D).
We give an intrinsic proof of this theorem below. Thus, for geodesically invariant distribu-
tions, the symmetric product plays the role that the Lie bracket plays for integrable distributions.
We use this result to interpret some terms obtained in the decomposition of the reduced geodesic
spray in Section 4.
Now, given a p-dimensional distribution D on an n-dimensional manifold M with a linear
connection, we say that a frame u ∈ Lx(M) is D-adapted if u|Rp is an isomorphism onto Dx. Let
L(M,D) be the collection of D-adapted frames. We observe that L(M,D) is invariant under
the subgroup of GL(n; R) consisting of those automorphisms which leave Rp invariant. It turns
out that L(M,D) is a subbundle of L(M) with structure group H given by
H =
{
A ∈ GL(n; R) |A =
(
a b
0 c
)
, a ∈ GL(p; R), b ∈ L(Rn−p, Rp), c ∈ GL(n− p; R)
}
.
We denote the bundle of D-adapted frames by L(M,D)(M,H) and the Lie algebra of H by h.
We have the following result.
Proposition 3.9. The distribution D is geodesically invariant if and only if, for each ξ ∈ Rp,
B(ξ ⊕ 0)|L(M,D) is a vector field on L(M,D).
Proof. We first prove the “if” statement. Suppose that B(ξ ⊕ 0) is a vector field on L(M,D)
and let c : R → L(M) be its integral curve passing through ū ∈ L(M,D). Then, we know that
x(t) := πM (c(t)) is the unique geodesic with the initial condition ūξ ∈ D. We must show that
ẋ(t) ∈ Dx(t) for all t. We have
ẋ(t) = TπM (B(ξ ⊕ 0)c(t)) = c(t)(ξ ⊕ 0).
Since B(ξ ⊕ 0) is a vector field on L(M,D), we must have c(t) ∈ L(M,D) for all t. Thus, we
have ẋ(t) ∈ Dx(t) for all t. The “only if” part of the statement can be proved by reversing this
argument. �
An immediate consequence of this result is the following.
Corollary 3.10. A distribution D is geodesically invariant if and only if the geodesic spray Z
is tangent to the submanifold D of TM .
We are now in a position to provide a proof of Theorem 3.8 using frame bundle geometry.
Proof of Theorem 3.8. (i) =⇒ (ii) Suppose that D is geodesically invariant, and let X1, X2 ∈
Γ(D). Then, we know that the corresponding functions fXi : L(M,D) → Rp ⊕ Rn−p, i = 1, 2,
take values in Rp. Also,
(Xi)h(u) = cj
iB(ej ⊕ 0)u, u ∈ L(M,D),
Geodesic Reduction via Frame Bundle Geometry 9
where cj
i are functions on L(M) and {ej}j=1,...,p is the standard basis for Rp. This is possible
since {B(ei)} form a basis for HuL(M). We have
f(∇X1
X2+∇X2
X1) = L(X1)hfX2 + L(X2)hfX1 = cj
1LB(ej⊕0)fX2 + ck
2LB(ek⊕0)fX1 .
Since fXi , i = 1, 2, are Rp-valued functions on L(M,D) and B(ej ⊕ 0)|L(M,D), j = 1, . . . , p, are
vector fields on L(M,D) because the distribution is assumed to be geodesically invariant, we
conclude that the function f(∇X1
X2+∇X2
X1) : L(M,D) → Rp⊕Rn−p takes its values in Rp. This
proves (ii).
(ii) =⇒ (iii) This follows directly from the definition of the symmetric product.
(iii) =⇒ (i) Assume that ∇XX ∈ Γ(D) for every X ∈ Γ(D). This implies that the function
LXhfX : L(M,D) → Rp⊕Rn−p takes values in Rp. Once again, we can write Xh = CiB(ei⊕0)
for some functions Ci. This implies that B(ei⊕0)|L(M,D) must be a vector field on L(M,D). �
The above result shows that it is possible to check for geodesic invariance by looking at vector
fields B(ei ⊕ 0) on the bundle L(M,D).
Theorem 3.8 and Proposition 3.6 suggest the following:
Corollary 3.11. Let D be a p-dimensional distribution on a manifold M with a linear connec-
tion ω. Let D̃ be the natural lift of D onto L(M) and L(M,D)(M,H) the bundle adapted to D.
The following statements are equivalent:
(i) D is geodesically invariant;
(ii) Sym(ω ⊗ θ) is an Rp-valued quadratic form on D̃|L(M,D);
(iii) ω(X̃(u)) ∈ h for all X̃ ∈ Γ(D̃), u ∈ L(M,D).
4 Geodesic reduction
In this section we consider the following setup. Let M(M/G,G) be a principal fiber bundle with
structure group G and let ∇ be a G-invariant affine connection on M . We study the “reduced”
geodesic spray corresponding to ∇ and use a principal connection to decompose it into various
components. These components correspond to geometric objects defined on M/G.
4.1 Invariant affine connections
Let M(M/G,G) be a principal fiber bundle with a G-invariant affine connection ∇ on M .
Choose a principal connection A on M(M/G,G). With this data, we can define an affine
connection ∇A on M/G as follows.
Proposition 4.1. Let ∇ and A be as above. Given vector fields X and Y ∈ Γ(T (M/G))
on M/G, the map ∇A : Γ(T (M/G))× Γ(T (M/G)) → Γ(T (M/G)) defined by
∇A
XY (x) = TπM/G∇XhY h(q), q ∈ M, πM/G(q) = x ∈ M/G (4.1)
is an affine connection on M/G.
Proof. It is easy to see that ∇A is well-defined. Given a smooth function f : M/G → R, we
define the lift f̃ : M → R by f̃ = π∗M/Gf . We have
∇A
X(fY )(x) = TπM/G∇Xh(f̃Y h)(q) = TπM/G
(
f̃(q)∇XhY h(q) + (Lxh f̃)(q) · Y h(q)
)
= f(x)∇A
XY (x) + (LXf)(x) · Y (x),
where the last part follows since the integral curves of Xh project to integral curves of X. The
fact that ∇A satisfies all the other properties of an affine connection is easily verified. �
10 A. Bhand
4.2 The reduced geodesic spray and its decomposition
In this section we carry out the reduction of the geodesic spray corresponding to an invariant
affine connections. Let M(M/G,G) be a principal fiber bundle with total space M and structure
group G. We denote the action of G on M by Φ : G × M → M and its tangent lift by
ΦT : G × TM → TM . The tangent bundle projection is denoted by τM : TM → M . We can
define a map [τM ]G : TM/G → M/G as follows:
[τM ]G([v]G) = [τM (v)]G, [v]G ∈ TM/G.
It is easy to see that [τM ]G : TM/G → M/G is a vector bundle. The adjoint bundle with g as the
fiber and M/G as the base space will be represented by g̃M/G. A typical element of g̃M/G will
be denoted by [x, ξ]G, where x ∈ M and ξ ∈ g. We shall also denote the tangent bundle of M/G
by τM/G : T (M/G) → M/G. If A is a principal connection on the bundle πM/G : M → M/G,
we can decompose the bundle TM/G into its horizontal and vertical parts [19].
Lemma 4.2. The map αA : TM/G → T (M/G)⊕ g̃M/G given by
αA([vx]G) = TπM/G(vx)⊕ [x,A(vx)]G, vx ∈ TxM, x ∈ M
is a vector bundle isomorphism.
We denote the g̃ component of αA by ρA : TM/G → g̃. This decomposition of TM/G is
A-dependent and we write TM/G ' T (M/G) ⊕A g̃M/G. The lifted action ΦT makes TM the
total space of a principal bundle over TM/G with structure group G. We denote the canonical
projection by πTM/G : TM → TM/G. Furthermore, G acts on TTM by the tangent lift of ΦT .
We denote by TτM : TTM/G → TM/G the map given by
TτM ([Wvx ]G) = [TτM (Wvx)]G, Wvx ∈ TvxTM.
It is easy to see that this map is well-defined. By Lemma 4.2, a principal connection  on
TM(TM/G, G) induces an isomorphism between bundles TTM/G and T (TM/G) ⊕ g̃TM/G
over TM/G. Thus, if A and  are chosen, we can consider an identification of TTM/G and
TT (M/G)⊕A T g̃M/G⊕Â g̃TM/G where we identify T (TM/G) and T (T (M/G))⊕A T g̃ using the
map TαA. The following Lemma will be useful in our decomposition of the geodesic spray.
Lemma 4.3. Given a principal connection A on M(M/G,G), the pullback  := τ∗MA is a prin-
cipal connection on TM(TM/G, G).
The connection  has the following useful property.
Corollary 4.4. If S : TM → TTM is a second-order vector field, then
ρÂ([S(vx)]G) = [vx, A(vx)]G ∈ g̃TM/G.
In other words, if we choose connections A and  on M(M/G,G) and TM(TM/G, G) re-
spectively, studying a second-order vector field such as the geodesic spray reduces to studying
the TT (M/G) ⊕ T g̃ components, since the g̃TM/G component is completely determined by A
itself. We now define the reduced geodesic spray.
Proposition 4.5. Let ω be a G-invariant linear connection on L(M) and ∇ the corresponding
connection on M . The map Z : TM/G → TTM/G given by
Z([vx]G) = [Z(vx)]G = [TuΦξB(ξ)u]G
is well-defined. We call Z the reduced geodesic spray.
Geodesic Reduction via Frame Bundle Geometry 11
Proof. G-invariance of ω implies the invariance of the standard horizontal vector fields. The
result now follows from the G-equivariance of the association map Φξ : L(M) → TM , ξ ∈ Rn. �
Now, since G acts on TTM via the lifted action, we can define a map TπTM/G : TTM/G →
T (T (M/G)) as follows:
TπTM/G[Wvx ]G = TπTM/G(Wvx), [Wvx ]G ∈ TTM/G.
This is well-defined since given any g ∈ G, we have πTM/G ◦ TΦT
g = πTM/G. By abuse of
notation, we shall use the maps TπTM/G and TπTM/G interchangeably. We have the following
result.
Proposition 4.6. Let SZ : T (M/G) → TT (M/G) be the map defined by
SZ(X̄) = T (TπM/G ◦ πTM/G)Z([X̄h(x)]G), X̄ ∈ T[x]G(M/G),
where X̄h is an invariant horizontal vector field that projects to X̄ at x ∈ M . The following
statements hold:
(i) SZ is a second-order vector field on T (M/G);
(ii) SZ(X̄) = X̄T (X̄)− vlftX̄TπM/G(∇X̄hX̄h), where, by abuse of notation, X̄ is a vector field
on M/G which has a value X̄ at [x]G ∈ M/G.
Proof. (i) We compute
TτM/GSZ(X̄) = T (τM/G ◦ TπM/G)Z
(
X̄h(x)
)
= TπM/GTτM
(
Z
(
X̄h(x)
))
= TπM/G
(
X̄h(x)
)
= X̄.
(ii) Let ΦX̄h
t and ΦX̄
t be the flows of X̄h and X̄ respectively. We have
TTπM/G
(
X̄h
)T (
X̄h(x)
)
=
d
dt
∣∣∣∣
t=0
(
TπM/G ◦ TΦXh
t
)(
X̄h(x)
)
=
d
dt
∣∣∣∣
t=0
T
(
πM/G ◦ ΦXh
t
)(
X̄h(x)
)
=
d
dt
∣∣∣∣
t=0
(
d
ds
∣∣∣∣
s=0
(
πM/G ◦ ΦXh
t
)(
ΦXh
s (x)
))
=
d
dt
∣∣∣∣
t=0
d
ds
∣∣∣∣
s=0
(
πM/G ◦ ΦXh
t+s(x)
)
=
d
dt
∣∣∣∣
t=0
d
ds
∣∣∣∣
s=0
(
ΦX
t+s([x]G
)
= X̄T (X̄).
Next, we look at
TTπM/GvlftX̄h(x)
(
∇X̄hX̄h(x)
)
=
d
dt
(
tTπM/G∇X̄hX̄h(x) + TπM/GX̄h(x)
)∣∣∣∣
t=0
=
d
dt
(
tTπM/G∇X̄hX̄h(x) + X̄([x]G)
)∣∣∣∣
t=0
= vlftX̄([x]G)TπM/G∇X̄hX̄h(x).
This gives us
SZ(X̄) = X̄T (X̄)− vlftX̄
(
TπM/G∇X̄hX̄h(x)
)
.
The result now following from Proposition 4.1. �
12 A. Bhand
The idea here is that we use principal connections A and τ∗MA on M(M/G,G) and
TM(TM/G, G), respectively, to write the reduced geodesic spray corresponding to an invariant
linear connection as a map from T (M/G) ⊕ g̃ to TT (M/G) ⊕ T g̃. The map SZ gives us one
component of this decomposition. From Proposition 3.4, we see that SZ is the geodesic spray of
the affine connection ∇A on M/G.
Next, we define a map PZ : g̃ → TT (M/G) as follows:
PZ([x, ξ]G) = TTπM/GTπTM/GZ([ξV
L (x)]G),
where ξV
L is the left-invariant vector field on M that satisfies ξV
L (x) = ξM (x). We must verify
that this is well-defined. To see this, notice that [g · x,Adgξ]G = [x, ξ]G. Next, we have
(Adgξ)M (g · x) =
d
dt
Φexp(Adgξ)t(g · x)
∣∣∣∣
t=0
=
d
dt
Φ
(
g(exp ξt)g−1, g · x
)∣∣∣∣
t=0
= TxΦgξM (x) = ξV
L (g · x).
Let us denote ξ̃ := [x, ξ]G. Using (3.5), we get
PZ(ξ̃) = TTπM/GZ
(
ξV
L (x)
)
= TTπM/G
((
ξV
L
)T (
ξV
L (x)
))
− TTπM/GvlftξV
L (x)
(
∇ξV
L
ξV
L (x)
)
= −vlft0
(
TπM/G∇ξV
L
ξV
L (x)
)
.
We write S (ξ̃, ξ̃) = (TπM/G∇ξV
L
ξV
L (x)). Since ∇ is G-invariant, this map is well-defined.
Next, we define RZ : T (M/G) → T g̃ by
RZ(X̄) = TρATπTM/GZ
(
X̄h(x)
)
.
Then, using (3.5), we calculate
TρATπTM/GZ
(
X̄h(x)
)
= TρATπTM/G
((
X̄h
)T (
X̄h(x)
)
− vlftX̄h(x)
(
∇X̄hX̄h(x)
))
.
Let us look at the first term on the right-hand side
TρATπTM/G
((
X̄h
)T (
X̄h(x)
))
= TρATπTM/G
d
dt
∣∣∣∣
t=0
TΦX̄h
t
(
X̄h(x)
)
=
d
dt
∣∣∣∣
t=0
ρA
(
[TΦX̄h
t
(
X̄h(x)
)
]G
)
= 0,
since TΦX̄h
t (X̄h(x)) is horizontal and ρA vanishes on horizontal vectors. Also,
TρATπTM/GvlftX̄h(x)
(
∇X̄hX̄h(x)
)
=
d
dt
∣∣∣∣
t=0
ρAπTM/G
(
t∇X̄hX̄h(x) + X̄h(x)
)
=
d
dt
∣∣∣∣
t=0
(
tρA ◦ πTM/G
(
∇X̄hX̄h(x)
)
+ ρA ◦ πTM/G
(
X̄h(x)
))
=
d
dt
∣∣∣∣
t=0
(
tρA ◦ πTM/G
(
∇X̄hX̄h(x)
)
+ 0
)
,
and thus we get
RZ(X̄) = −vlft0
(
ρA ◦ πTM/G
(
∇X̄hX̄h(x)
))
.
If HM is geodesically invariant, then ∇X̄hX̄h is horizontal, and thus RZ = 0.
Geodesic Reduction via Frame Bundle Geometry 13
Finally, we define UZ : g̃ → T g̃ by
UZ(ξ̃) = TρATπTM/GZ
(
ξV
L (x)
)
,
and a calculation similar to the one performed above shows that
UZ(ξ̃) = −vlftξρA
(
πTM/G
(
∇ξV
L
ξV
L (x)
))
.
The following lemma is useful.
Lemma 4.7. The map ∇̃A : Γ(T (M/G))× Γ(g̃) → Γ(g̃) given by
∇̃A
X̄ ξ̃([x]G) = ρAπTM/G
(〈
X̄h : ξV
L
〉
(x)
)
, [x]G ∈ (M/G)
defines a vector bundle connection on the bundle g̃.
Proof. Let f : M/G → R be a differentiable function. Define fh : M → R by fh = π∗M/Gf .
Therefore, (fX̄)h = fhX̄h. We compute
∇̃A
fX̄ ξ̃ = ρAπTM/G
(〈
fhX̄h : ξV
L
〉)
= ρAπTM/G
(
fh∇X̄hξV
L + fh∇ξV
L
X̄h +
(
LξV
L
fh
)
X̄h
)
= f∇̃A
X̄ ξ̃,
since (LξV
L
fh)X̄h = 0. The property ∇̃A
X̄
f ξ̃ = f∇̃A
X̄
ξ̃ + (LX̄f)ξ̃ can be proved similarly. �
We now state the main result of this section.
Theorem 4.8. Let Zh : T (M/G)⊕ g̃ → TT (M/G) be the map defined by
Zh(X̄ ⊕ ξ̃) = TTπM/GZ
[
X̄h(x) + ξV
L (x)
]
G
,
where X̄h is an invariant horizontal vector field that projects to X̄ at x ∈ M , and ξV
L is the
left-invariant vertical vector field with value ξM (x) at x ∈ M .
Let Zv : T (M/G)⊕ g̃ → T g̃ be the map defined by
Zv(X̄ ⊕ ξ̃) = TρAZ
([
X̄h(x) + ξV
L (x)
]
G
)
,
where X̄h and ξV
L are defined as above. The following statements hold:
(i) Zh(X̄ ⊕ ξ̃) = SZ(X̄)− vlftX̄S (ξ̃, ξ̃)− vlftX̄
(
TπM/G
〈
X̄h : ξV
L
〉)
;
(ii) Zv(X̃ ⊕ ξ̃) = RZ(X̄) + UZ(ξ̃)− vlftξ
(
∇̃A
X̄
ξ̃([x]G)
)
.
Proof. Let us compute
TTπM/GZ
(
X̄h(x) + ξV
L (x)
)
= TTπM/G
((
X̄h + ξV
L
)T (
X̄h + ξV
L (x)
))
− vlftX̄h(x)
(
TπM/G∇X̄h+ξV
L
(Xh + ξV
L )
)
= TTπM/G
(
X̄h
)T (ξM (x)) + X̄T (X̄([x]G)− vlftX̄h(x)
(
TπM/G∇X̄hX̄h
)
− vlftX̄h(x)
(
S (ξ̃, ξ̃)
)
− vlftX̄h(x)
(
TπM/G
〈
X̄h : ξV
L
〉)
= TTπM/G
(
X̄h
)T (
ξV
L (x)
)
+ SZ(X̄)− vlftX̄h(x)
(
S (ξ̃, ξ̃)
)
− vlftX̄h(x)
(
TπM/G
〈
X̄h : ξV
L
〉
(x)
)
.
We also have
TTπM/G
(
X̄h + ξV
L
)T (
ξV
L (x)
)
=
d
dt
∣∣∣∣
t=0
TπM/GξV
L
(
ΦX̄h
t (x)
)
= 0.
This gives us the first part. Part (ii) follows from a similar computation. �
14 A. Bhand
Remark 4.9. The fact that the right-hand sides of Zh and Zv respectively are independent of
the extensions follows from G-invariance of ω and the definition of Z.
Remark 4.10. The decomposition of the reduced geodesic spray into horizontal and vertical
parts given in Theorem 4.8 is similar to the decomposition of second-order systems in Crampin
and Mestdag [13], particularly in the case of an affine spray. We discuss the Riemannian case
in Section 5.1.
5 Discussion
The horizontal part of the reduced geodesic spray consists of three terms. The map SZ is
a second-order vector field on T (M/G). The term S (ξ̃, ξ̃) can be interpreted in the following
manner. Recall that the second fundamental form corresponding to the vertical distribution is
a map S : Γ(V M)× Γ(V M) → HM defined by
S(vx, wx) = hor (∇XY ) , vx, wx ∈ VxM,
where X and Y are extensions of vx and wx respectively. In view of this, we have
S(ξM (x), ξM (x)) =
(
S (ξ̃, ξ̃)
)h(x).
Now, the vertical distribution V M is geodesically invariant if and only if S is skew-symmetric.
Hence, if V M is geodesically invariant, we have S (ξ̃, ξ̃) = 0.
5.1 The Riemannian case
The last term in the horizontal part of the reduced geodesic spray is related to the curvature
of the horizontal distribution, at least in the case when M is a Riemannian manifold with
an invariant Riemannian metric, the chosen affine connection is the Levi-Civita connection
corresponding to this metric, and A is the mechanical connection as we show below.
Let (M,k) be a Riemannian manifold and G be a Lie group that acts freely and properly
on G, so that πM/G : M → M/G is a principal bundle. Suppose that the Riemannian metric k
is invariant under G. The mechanical connection corresponding to k is a principal connection on
πM/G : M → M/G determined by the condition that the horizontal subbundle is orthogonal to
the vertical subbundle V M with respect to the metric. We denote by A the connection one-form
corresponding to this connection. We also let ∇ be the Levi-Civita connection corresponding
to k.
Lemma 5.1. The following holds
k
(〈
X̄h : ξV
L
〉
(x), Ȳ h(x)
)
= k
((
BA
(
X̄h(x), Ȳ h(x)
))
M
, ξV
L (x)
)
,
where X̄h and Ȳ h are invariant horizontal vector fields on M , and BA is the curvature form
corresponding to A.
Proof. Recall that if X, Y and Z are vector fields on M , the Koszul formula is given by
2k(∇XY, Z) = LX(k(Y, Z)) + LY (k(X, Z))−LZ(k(X, Y )) + k([X, Y ], Z)
− k([X, Z], Y ])− k([Y, Z], X).
We therefore have (using the Koszul formula twice and adding the two results)
2k
(〈
X̄h : ξV
L
〉
(x), Ȳ h(x)
)
= 2LX̄h
(
k
(
Ȳ h(x), ξV
L (x)
))
+ 2LξV
L
(
k
(
X̄h(x), Ȳ h(x)
))
− 2LȲ h
(
k
(
X̄h(x), ξV
L (x)
))
− 2k
([
X̄h, Ȳ h
]
(x), ξV
L (x)
)
− 2k
([
ξV
L , Ȳ h
]
(x), X̄h(x)
)
.
Geodesic Reduction via Frame Bundle Geometry 15
Now, the first and the third terms respectively on the right-hand side are clearly zero (by
the definition of the mechanical connection). The second term is zero since the function
k
(
x̄h(x), Ȳ h(x)
)
is constant along the invariant vertical vector field ξV
L . The fifth term is
also zero since the Lie bracket [ξV
L , Ȳ h] is a vertical vector field. Thus, we get
k
(〈
X̄h : ξV
L
〉
(x), Ȳ h(x)
)
= k
([
X̄h, Ȳ h
]
(x), ξV
L (x)
)
.
By the Cartan structure formula, we have[
X̄h, Ȳ h
]
= [X̄, Ȳ ]h −
(
BA
(
X̄h, Ȳ h
))
M
(x).
Therefore,
k
(〈
X̄h : ξV
L
〉
(x), Ȳ h(x)
)
= k
((
BA
(
X̄h(x), Ȳ h(x)
))
M
, ξV
L (x)
)
. �
The vertical part of the reduced geodesic spray consists of the map RZ which vanishes iden-
tically if the horizontal distribution corresponding to the principal connection A is geodesically
invariant, and can be thought of as the fundamental form corresponding to the horizontal dis-
tribution. Lewis [17] has shown that if both HM and V M are geodesically invariant, then
the corresponding linear connection restricts to the subbundle L(M,A). The term UZ(ξ̃) is
essentially the Euler–Poincaré term, and the last term corresponds to a connection on g̃.
5.2 Forces in mechanics
As mentioned in the Introduction, our motivation for studying reduction in the affine connection
setup comes from mechanics. In this sense, studying the geodesic spray corresponds to looking
at mechanical systems with no external forces (in other words, the dynamics are given by
the geodesic equation). It is worth considering the case in which forces are present as many
important examples in mechanics fall in this class. In the following, we consider the so-called
simple mechanical systems [1, 5, 21]. A simple mechanical system is a triple (M,k, V ) where
(M,k) is Riemannian manifold and V : M → R is a smooth function (called the potential
function). The gradient of V is a vector field on M defined by
gradV (x) = k#(dV (x)),
where k# : T ∗M → TM is a vector bundle isomorphism over M induced by the metric k. The
dynamics of such a system are given by
∇c′(t)c
′(t) = −(gradV )(c(t)), c(t) ∈ M, (5.1)
where ∇ is the Levi-Civita connection on M . Equivalently, one can study the following equation
on TM [1]:
v′(t) = Z(v(t))− vlftv(t) (gradV (c(t))) , v(t) ∈ TM, τM (v(t)) = c(t). (5.2)
In the unforced case dealt with in this paper, we study the geodesic spray because its integral
curves project to geodesics on M . Similarly, we can study the second-order vector field
Z̃(v) = Z(v)− vlftv (gradV (τM (v))) ,
which, by Proposition 3.4, can be written as
Z̃(v) = XT (v)− vlftv (∇XvXv + gradV (τM (v))) .
If ∇, k and V are G-invariant, we can study the reduction of Z̃ using our methodology. Even
though we consider a potential force here, a general force F can be incorporated in this picture
by essentially replacing gradV with k#(F ) in (5.1).
16 A. Bhand
5.3 Generalized connections
In our investigation, we have considered a G-invariant affine connection∇ on a manifold M along
with a principal connection A on πM/G : M → M/G and used it to define an A-dependent affine
connection ∇A on M/G. Equivalently, and perhaps more naturally, ∇ induces a connection
on the vector bundle [τM ]G : TM/G → M/G. Since the principal connection A provides
a decomposition of TM/G, we can also recover ∇A in this manner.
Furthermore, we can consider a generalized connection [8, 11] on the vector bundle [τM ]G :
TM/G → M/G and explore how our reduction procedure can be applied to this more general
situation.
5.4 Nonholonomic systems with symmetry
Roughly speaking, nonholonomic systems are mechanical systems with velocities constrained
to lie in a given non-integrable distribution. Following the fundamental paper of Koiller [16]
there has been a lot of interest in studying symmetries and reduction of nonholonomic systems
[2, 4, 5, 7, 12, 17, 22]. In this section we outline how these systems can be studied in our
framework. Let (M,k) be a Riemannian manifold with a G-invariant Riemannian metric. Let D
be a smooth, non-integrable, G-invariant distribution on M and D⊥ the orthogonal complement
with respect to the metric k. Let ∇ be the Levi-Civita affine connection associated with k. The
Lagrange–d’Alembert principle allows us to conclude that the constrained geodesics c(t) ∈ M
satisfy [17]
∇c′(t)c
′(t) ∈ D⊥
c(t), c′(t) ∈ Dc(t).
Sometimes these conditions are written as
∇c′(t)c
′(t) = λ(c(t)), P⊥(c′(t)) = 0,
where λ is a section of D⊥ and P⊥ : TM → TM is the projection onto D⊥. It can be shown
that the trajectories c : R → M satisfying the constraints are actually geodesics of an affine
connection ∇̃ defined by ∇̃XY = ∇XY + (∇XP⊥)(Y ). Note that, in general, the connection ∇̃
(sometimes called a constrained connection) will not be Levi-Civita. We can use our approach to
study the geodesic spray of the constrained connection in the presence of a principal connection
on πM/G : M → M/G. In such a case, the picture gets more complicated since a decomposition
of TTM/G, and that of the geodesic spray, will depend on the distribution D, and we hope to
address this problem in subsequent work. This procedure is related to the reduction of “external”
symmetries of a generalized G-Chaplygin system [14].
Acknowledgements
I would like to thank my thesis supervisor Dr. Andrew Lewis for his constant guidance and sup-
port. This work would not have materialized without the many invaluable discussions I have had
with him over the years. The author also thanks the anonymous referees for their constructive
comments on a previous version of this paper.
References
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Geodesic Reduction via Frame Bundle Geometry 17
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http://dx.doi.org/10.1016/S0034-4877(98)80002-5
1 Introduction
2 Linear connections
3 The geodesic spray of an affine connection
3.1 Tangent and vertical lifts
3.2 Decomposition of the geodesic spray
3.3 Geodesic invariance
4 Geodesic reduction
4.1 Invariant affine connections
4.2 The reduced geodesic spray and its decomposition
5 Discussion
5.1 The Riemannian case
5.2 Forces in mechanics
5.3 Generalized connections
5.4 Nonholonomic systems with symmetry
References
|
| id | nasplib_isofts_kiev_ua-123456789-146313 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:10:08Z |
| publishDate | 2010 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bhand, A. 2019-02-08T20:28:17Z 2019-02-08T20:28:17Z 2010 Geodesic Reduction via Frame Bundle Geometry / A. Bhand // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53B05; 53C05; 53C22; 58D19 DOI:10.3842/SIGMA.2010.020 https://nasplib.isofts.kiev.ua/handle/123456789/146313 A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation. I would like to thank my thesis supervisor Dr. Andrew Lewis for his constant guidance and support. This work would not have materialized without the many invaluable discussions I have had with him over the years. The author also thanks the anonymous referees for their constructive comments on a previous version of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Geodesic Reduction via Frame Bundle Geometry Article published earlier |
| spellingShingle | Geodesic Reduction via Frame Bundle Geometry Bhand, A. |
| title | Geodesic Reduction via Frame Bundle Geometry |
| title_full | Geodesic Reduction via Frame Bundle Geometry |
| title_fullStr | Geodesic Reduction via Frame Bundle Geometry |
| title_full_unstemmed | Geodesic Reduction via Frame Bundle Geometry |
| title_short | Geodesic Reduction via Frame Bundle Geometry |
| title_sort | geodesic reduction via frame bundle geometry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146313 |
| work_keys_str_mv | AT bhanda geodesicreductionviaframebundlegeometry |