=Paper=
{{Paper
|id=Vol-2098/paper19
|storemode=property
|title=The Schouten Curvature for a Nonholonomic
Distribution in Sub-Riemannian Geometry and
Jacobi Fields
|pdfUrl=https://ceur-ws.org/Vol-2098/paper19.pdf
|volume=Vol-2098
|authors=Victor R. Krym
}}
==The Schouten Curvature for a Nonholonomic
Distribution in Sub-Riemannian Geometry and
Jacobi Fields==
https://ceur-ws.org/Vol-2098/paper19.pdf
The Schouten Curvature for a Nonholonomic
Distribution in Sub-Riemannian Geometry and
Jacobi Fields
Victor R. Krym
DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge
CB3 0WA, United Kingdom
vkrym12@rambler.ru
Abstract. The paper shows that if the distribution is defined on a man-
ifold with the special smooth structure and does not depend on the ver-
tical coordinates, then the Schouten curvature tensor coincides with the
Riemannian curvature tensor. The Schouten curvature tensor is used to
write the Jacobi equation for the distribution. This leads to studies on
second-order optimality conditions for the horizontal geodesics in sub-
Riemannian geometry. Conjugate points are defined by the solutions of
the Jacobi equation. If a geodesic passed a point conjugated with its
beginning then this geodesic ceases to be optimal.
Keywords: Sub-Riemannian geometry · Nonholonomic distributions ·
Schouten curvature · Jacobi fields
1 Introduction
Distribution on a smooth manifold N is a family of subspaces A(x) ⊂ Tx N ,
x ∈ N . General theory of the variational calculus with nonholonomic restrictions
ϕ(t, x, ẋ) = 0 was published by G.A. Bliss [2]. If the restrictions are linear for
velocities, ωx (ẋ) = 0, where ω is a 1-form, we get distributions [25, 26].
The Romanian mathematician Vranceanu first introduced the term of the
nonholonomic structure on a Riemannian manifold in 1928 [27]. The Dutch
mathematician Schouten defined the connection and appropriate curvature ten-
sor for horizontal vector fields on a distribution in [23, 22]. The soviet geometri-
cian Vagner extended this construction and built the general curvature tensor
compatible with the Schouten tensor satisfying common requirements for the
curvature in 1937, Kazan [24]. These results were published by Gorbatenko,
Tomsk, in 1985 [6] with modern notations.
The Schouten curvature tensor is different from the Riemannian curvature.
In this paper we prove that if the distribution is defined on a manifold with the
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org
214 V. R. Krym
special smooth structure and does not depend on the “vertical” coordinates then
these two tensors are identical. This curvature tensor is necessary to write the
Jacobi equation for a distribution.
We study also second-order optimality conditions for the horizontal geodesics
in sub-Riemannian geometry. If a geodesic passed a point conjugated with its
beginning then this geodesic ceases to be optimal. Conjugate points are defined
by means of the solutions of the Jacobi equation. In this paper we discuss several
examples of conjugate points in sub-Riemannian geometry.
2 Variations and Equations of Variations
Let γ : [t0 , T ] → N be a C 1 -smooth horizontal path, ω(γ 0 ) = 0, where ω is a 1-
form. Horizontal variation of γ is a 1-parametric family of maps σ(·, τ ) : [t1 , t2 ] →
N , |τ | < ε, if there are continuous vector fields X = ∂σ , Y = ∂σ , the central
∂t ∂τ
line is just σ(t, 0) = γ(t) and the field X is horizontal, X(t, τ ) ∈ A(σ(t, τ )) and
2 2
there are continuous second derivatives ∂ σ , ∂ σ for all allowed t and τ .
∂t∂τ ∂τ ∂t
Since the horizontality condition ω α ( ∂σ ) = 0 is fulfilled as an identity we
∂t
n
∂ωkα ∂σ j ∂σ k P n 2 k
ωkα ∂ σ = 0. At τ = 0
P
can differentiate it for τ . We get j ∂τ ∂t +
j,k=1 ∂x k=1
∂t∂τ
we obtain the equations of variations along γ:
n n
X dY k X ∂ω α
k 0k j
ωkα + γ Y = 0, α = m+1, . . ., n. (1)
k=1
dt ∂xj
j,k=1
The vector field Y (·, 0) along γ is denoted as Y . These equations are of the form
Φα (Y 0 , Y ) = 0, α = m+1, . . ., n. This is a system of n − m differential equations.
Since the rank of the matrix (ωkα ) is n − m, the horizontal projection of the
field Y is arbitrary and the vertical components Y α are defined by the initial
conditions.
3 The Accessory Problem of G.A. Bliss (Example)
3.1 The Lagrange Problem
Let us consider the 2-dimensional distribution A on R3 defined by the 1-form
RT
ω = x2 dx1 + dx3 . The energy functional is J(x(·)) = 12 0 (ẋ1 )2 + (ẋ2 )2 dt, the
metric tensor for this distribution is the identity matrix. The Lagrangian of the
variational problem is L = 21 (ẋ1 )2 + (ẋ2 )2 + lω(γ 0 ). The horizontal geodesics
starting from the origin in R3 for l 6= 0 are
1
x (t) = 1l −v2 + v2 cos lt + v1 sin lt ,
x2 (t) = 1l v1 − v1 cos lt + v2 sin lt ,
x3 (t) = 4l12 2lt(v12 + v22 ) + 2v1 v2 − 4v1 v2 cos lt − 4v12 sin lt+ (2)
+2v1 v2 cos 2lt + (v12 − v22 ) sin 2lt .
The Schouten Curvature for a Nonholonomic Distribution 215
Note that ẋ1 (0) = v1 and ẋ2 (0) = v2 . These geodesics are shown at Fig. 1 for
Fig. 1. Horizontal geodesics.
l = −1. These geodesics are optimal for the interval [0, 2π/|l|). If |lt| > 2π, then
the geodesic ceased to be optimal.
3.2 The Accessory Problem
RT
Let us consider the minimization problem for the functional t0 f (t, x, ẋ) dt
with the nonholonomic constrains ϕ(t, x, ẋ) = 0. Assume that the matrix
∂ϕα
α=m+1,...,n, has the rank n−m for all (t, x, ẋ). Let L(t, x, ẋ, l) = f (t, x, ẋ)+
∂ ẋk k=1,...,n
n
lα ϕα (t, x, ẋ). The second variation of the functional
P
α=m+1
Z T (τ )
J(τ ) = L(t, σ(t, τ ), σt0 (t, τ ), l) dt (3)
t0 (τ )
is as following:
n n
!
2 d2 J 0
T X ∂L k X ∂L k T dL T
δ J= 2
= L(t, x, ẋ)ξτ +2 k
η + k
η̇ ξ + ξ 2 +
dτ τ =0 t0 ∂x ∂ ẋ t0 dt t0
k=1 k=1
n
X ∂L ∂η k T T X n
∂2L i j ∂2L ∂2L i j
Z
+ + i j
η̇ η̇ + 2 i j η i η̇ j + η η dt,
k=1
∂ ẋk ∂τ
t0 t0 i,j=1 ∂ ẋ ∂ ẋ ∂x ∂ ẋ ∂xi ∂xj
(4)
216 V. R. Krym
where ξ(t0 ) = dt dT 0 d2 t0 0 d2 T ∂σ
dτ , ξ(T ) = dτ , ξτ (t0 ) = dτ 2 , ξτ (T ) = dτ 2 , η = ∂τ at τ = 0.
0
This functional should be minimized for variables η satisfying the equations of
variations along γ [2]. Since the distribution A is defined by the 1-form ω =
x2 dx1 + dx3 in our example, the constrains (1) in the accessory problem are
Φ ≡ η̇ 3 + x2 η̇ 1 + η 2 ẋ1 = 0. (5)
This is the equation of variations along γ. Since the Lagrangian in our example
is L = 12 (ẋ1 )2 + (ẋ2 )2 + l(x2 ẋ1 + ẋ3 ), the Lagrangian for the accessory problem
is Ω = 12 (η̇ 1 )2 + (η̇ 2 )2 + lη 2 η̇ 1 + λ(η̇ 3 + x2 η̇ 1 + η 2 ẋ1 ). The generalized momenta
p1 = η̇ 1 + lη 2 + λx2 , p2 = η̇ 2 and p3 = λ. The generalized forces f1 = 0,
f2 = lη̇ 1 + λẋ1 and f3 = 0. The Euler—Lagrange equations for the accessory
problem are η̈ 1 +lη̇ 2 +λẋ2 = 0, η̈ 2 = lη̇ 1 +λẋ1 and λ̇ = 0. Applying the geodesics
(2) we get
1
η̈ + lη̇ 2 + λ v1 sin lt + v2 cos lt = 0
η̈ 2 − lη̇ 1 − λ v cos lt − v sin lt = 0
1 2
3 1
(6)
η̇ + l v 1 − v 1 cos lt + v 2 sin lt η̇ 1 + v1 cos lt − v2 sin lt η 2 = 0
λ̇ = 0.
This is the system of linear homogeneous differential equations for the variables
η, λ. To find points conjugate with the starting point t0 = 0 one should find
the solutions of these equations which are zero η(0) = 0 at the initial point but
η 0 (0) 6= 0. The solutions have the form
(η 1 , η 2 , η 3 )T = P (a1 , a2 , λ)T , (7)
where a1 , a2 , λ are constants and P is the following part of the fundamental
matrix of the system (6):
sin lt cos lt − 1 (v1 lt − v2 ) cos lt − (v1 + v2 lt) sin lt + v2
l l l2
1 − cos lt sin lt (v1 lt − v2 ) sin lt + (v1 + v2 lt) cos lt − v1
P =
l l l2 .
2 2 2 2 2
2v1 lt−4v1 sin lt+ 2v2 lt−v2 sin 2lt+ − (v1 +v2 )lt−4v1 sin lt+(v1 −v2 +2v1 v2 lt) sin 2lt+
+v1 cos 2lt+v1 − +2(v1 lt−2v2 )v1 cos lt−2v1 v2 lt sin lt+2v
1 v2 +
+v sin 2lt−2v cos lt+
1 2
2 2
+v2 cos 2lt+v2 −2v1 cos lt +(2v1 v2 +(v2 −v1 )lt) cos 2lt
2l2 2l2 2l3
If t1 is a point conjugated with the initial point then there is the solution η of (6)
satisfying η(t1 ) = 0. To find such a solution we should calculate the determinant
lt lt lt
det P = −2t lt cos − 2 sin sin( )(v12 + v22 )/l4 . (8)
2 2 2
Since det P (t1 ) = 0 we get two series of conjugate points. For the conjugate
points of first series sin lt2 = 0 and tk = 2πk/l, k ∈ Z. The first conjugate point
of this series is t1 = 2π/l (Fig. 1). The appropriate Jacobi field is defined by (7)
with the parameters
a1 = v2 , a2 = −v1 , λ = 0. (9)
The Schouten Curvature for a Nonholonomic Distribution 217
For the conjugate points of second series lt cos lt2 − 2 sin lt2 = 0. Approximately
tn ≈ ±(π + 2πn)/l, n ∈ N. The first conjugate point of this series is t1 ≈ 8.99/l
(Fig. 2). The appropriate Jacobi field is defined by (7) with the parameters
a1 ≈ 4.49v2 /l, a2 ≈ −4.49v1 /l, λ = 1. (10)
Fig. 2. Conjugate point of second series t1 ≈ 8.99.
The equations (2) define the exponential map for the geodesics with the
initial velocity vector v = (v1 , v2 ) and Lagrange multiplier l: x(t) = expl0 (tv).
This exponential map was studied in [9] and its differential was found. Let us
consider the family of paths
y(t, τ ) = expl+τ
0
µ
(t(v + τ b)), (11)
where b = (b1 , b2 ). This is the variation of the geodesic x(·). Let us find the
derivatives
∂y1
= − (lµt sin lt + µ cos lt − µ)v2 + (µ sin lt − lµt cos lt)v1 −
∂τ τ =0
b1 l sin lt − b2 l cos lt + b2 l /l2 ,
∂y2
= − (µ sin lt − lµt cos lt)v2 + (−lµt sin lt − µ cos lt + µ)v1 −
∂τ τ =0
b2 l sin lt + b1 l cos lt − b1 l /l2 ,
218 V. R. Krym
∂y3
= (µ sin 2lt − lµt cos 2lt − lµt)v22 +
∂τ τ =0
((−2lµt sin 2lt − 2µ cos 2lt + 2lµt sin lt + 4µ cos lt − 2µ)v1 −
b2 l sin 2lt + b1 l cos 2lt − 2b1 l cos lt + 2b2 l2 t + b1 l)v2 +
(−µ sin 2lt + lµt cos 2lt + 4µ sin lt − 2lµt cos lt − lµt)v12 +
(b1 l sin 2lt + b2 l cos 2lt − 4b1 l sin lt − 2b2 l cos lt + 2b1 l2 t + b2 l)v1 /(2l3 ).
The rank of exponential geodesic map is not its maximum rank at a conjugate
∂y
point. If t1 is a point conjugated with the initial point then (t ) = 0 for
∂τ τ =0 1
some µ and b. For the conjugate points of first series this is
(2πµv1 )/l2 = 0
(2πµv2 )/l2 = 0 (12)
(2π(µ(v12 + v22 ) − l(b2 v2 + b1 v1 )))/l3 = 0.
Therefore µ = 0 and b2 = −b1 v1 /v2 . This means that the Lagrange multiplier l
is fixed and b⊥v matching (9).
∂y
For the conjugate points of second series the equation (t) = 0 should
∂τ τ =0
be solved numerically. For the point t1 ≈ 8.99/l the solution is b1 ≈ 4.49v2 µ/l,
b2 ≈ −4.49v1 µ/l, µ 6= 0 matching (10). With this solution in mind we can draw
the geodesic variation with these parameters. This variation is shown on Fig. 2
for l = 1. The endpoints of the geodesics are close to each other at t1 ≈ 8.99/l
as should be expected for a conjugate point.
4 The Schouten Curvature and the Schouten–Vranceanu
Connection
Let us consider the distribution A of dimension m on a smooth manifold N
of dimension n. The coordinates xk , k = 1, . . ., n, on an open and sufficiently
small domain U ⊂ N can be chosen so as to maximize the projection of the
distribution A on the first m coordinates. Then the basis of the distribution A
can be chosen as
n
X
ek = ∂k − Aα
k ∂α , k = 1, . . ., m, (13)
α=m+1
where ∂k = ∂ k are the coordinate vector fields. The functions Aα k will be called
∂x
the potentials of the distribution. We assume that they are C 1 -smooth. The
m
distribution A can be defined also by differential 1-forms ω α = Aα s α
P
s dx + dx ,
s=1
α = m+1, . . ., n. Here and further Latin indexes are in the range 1, . . ., m and
m
ckij ek +
P
Greek indexes are in the range m+1, . . ., n. The Lie brackets [ei , ej ] =
k=1
The Schouten Curvature for a Nonholonomic Distribution 219
n
cα
P
ij ∂α . According to the choice (13) the only non-zero components are
α=m+1
Fijα = cα α
ij . The tensor Fij is the tensions tensor.
For any point x ∈ N the quadratic form h·, ·ix is defined on A(x). The metric
tensor in the basis (13) is
gij (x) = hei , ej ix . (14)
To define covariant differentiation ∇ on the distribution, we must introduce a
symmetric Riemannian connection. The property of being Riemannian is defined
in a standard way:
XhY, Zi = h∇X Y, Zi + hY, ∇X Zi, (15)
while the symmetry condition must be modified as
∇X Y − ∇Y X = pr([X, Y ]), (16)
m
ek ⊗ dxk is the (horizontal) projection on the distribution. To
P
where pr =
k=1
make this projection invariant with respect to transformations of coordinates, we
must impose the following constraints on the smooth structure of the manifold:
∂xk = 0, k = 1, . . ., m, α = m+1, . . ., n, and ∂xβ is the identity
∂y α ∂y α α,β=m+1,...,n
α
matrix [13, 14, 16]. The coordinates x will be called verticals. The differentials
of the transfer maps hU ◦ h−1 n
V for all charts hU : U → R , U, V ⊂ N , admissible
in this smooth structure are the block matrices
∂xi
α
∗
∂x ..
∂y j ∂y j = ?
. (17)
∂xi ∂xα
∗
0 . . . 0 In−m
∂y β ∂y β
Since these matrices form a group our definition is correct. The basis vectors
(13) change just as the coordinate basis of a manifold of dimension m: ei =
Pm ∂y s
s=1 ∂xi ẽs . The potentials of the distribution are also subject to the gauge
s
transformation: Ãα
Pm α ∂x ∂xα
j = s=1 As ∂y j + ∂y j .
Since the connection ∇ is Riemannian and pr-symmetric (15), (16) we get
2h∇X Y, Zi = (XhY, Zi − hY, pr[X, Z]i) + (Y hZ, Xi − hX, pr[Y, Z]i)−
− (ZhX, Y i − hZ, pr[X, Y ]i). (18)
Due to (13) pr[ei , ej ] = 0. Let us assume that the metric tensor of the distribution
does not depend on the vertical coordinates xα . Then Pthe equation (18) matches
m
the Levi-Civita connection on a manifold: ∇ei ej = k=1 Γijk ek and
m
1 X sk
Γijk =
g ∂i gjs + ∂j gis − ∂s gij . (19)
2 s=1
220 V. R. Krym
Theorem 1. For a distribution of dimension m on a manifold with the smooth
structure (17) the vectors of the basis (13) change over the coordinate transfor-
mations just as the coordinate basis of a manifold of the same dimension. If the
distribution and its metric tensor do not depend on the vertical coordinates then
the Riemannian and pr-symmetric connection ∇ matches the Riemannian and
symmetric connection of a manifold with metric (14).
For any point x ∈ N and any three vectors u, v, w ∈ A(x) the curvature map
of distribution A at point x is defined by the Schouten tensor [23, 22, 6]
R(u, v)w = ∇ũ ∇ṽ w̃ − ∇ṽ ∇ũ w̃ − ∇pr[ũ,ṽ] w̃ − pr (1 − pr)[ũ, ṽ], w̃ , (20)
where ũ, ṽ, w̃ are horizontal vector fields on a neighbourhood of x such that
ũ(x) = u, ṽ(x) = v, w̃(x) = w. The curvature does not depend on the way of ex-
pansion of u, v, w to vector fields. Assume that w̃ does not depend on vertical co-
ordinates xα and consider R(ei , ej )ek for the basis (13). The horizontal projection
of the Lie derivative on the vertical vector field is zero: pr (1−pr)[ei , ej ], ek = 0,
because ek does not depend on xα and the result of differentiation of the Lie
bracket [ei , ej ] on ek has the zero horizontal projection. Therefore the equation
(20) can be simplified: R(ei , ej )ek = ∇ei ∇ej ek − ∇ej ∇ei ek − ∇pr[ei ,ej ] ek . The
term ∇pr[ei ,ej ] ek = 0 since pr[ei , ej ] = 0. The corresponding term in Riemannian
geometry is also zero for any coordinate vector fields since [∂i , ∂j ] = 0. Final
equation for the curvature of distribution in the basis (13) matches the equation
for the curvature of a manifold: Rijkl = hR(ek , el )ej , ei i and
m
X
Ri jkl = ∂k Γlji − ∂l Γkj
i
Γljs Γks
i s
Γlsi .
+ − Γkj (21)
s=1
Theorem 2. Let the distribution be defined on a manifold with the smooth struc-
ture (17). If the distribution and its metric tensor do not depend on the vertical
coordinates then its Schouten tensor matches the Riemannian curvature tensor
of a manifold with metric (14).
“Look inside”: we cannot distinguish a distribution from a manifold locally. We
know that a distribution is involved in our physics because the equations of
motion of a charged particle are just the equations of horizontal geodesics [8,
15]. The potentials Aj for a 4-dimensional distribution on a 5-manifold are just
the 4-potentials of the electromagnetic field. The existence of these potentials is
confirmed by the Aharonov—Bohm effect [1]. The Maxwell equations and the
Dirac equation use this 4-potential and there is a problem how to find soliton-like
solution for the 4-potential satisfying these equations [4, 5]. If the distribution
and its metric tensor do not depend on the vertical coordinates we say that the
distribution satisfies the cyclicity condition.
5 The Equations of Geodesics
Let the distribution A on a manifold N n be defined by the differential 1-forms
m
ωα = Aα s α
P
s dx + dx , α = m+1, . . ., n. The geodesics equations for a distribu-
s=1
The Schouten Curvature for a Nonholonomic Distribution 221
tion with the cyclicity condition are [10]
n
Dγ 0 X
a0 + lα F̂ α γ 0 = 0, (22)
dt α=m+1
D
where dt – covariant derivative, (a0 , l) – the Lagrange multipliers, which cannot
be altogether zero. A geodesic γ is called normal (or regular), iff there is the
only one set of multipliers (a0 , l) with a0 = 1 for γ. Operator F̂ α is the tensions
tensor Fijα = ∂j Aα α
i − ∂i Aj with the second index raised by the inverse metric
tensor of the distribution.
Example 1. Let the 2-dimensional distribution
in R3 be defined by ω =
0 −2y
−y 2 dx+dz. The tensions tensor F = . Non-trivial abnormal geodesics
2y 0
appear at y = 0. It can be any absolutely continuous function x = x(t) with
z = const due to ω(γ 0 ) = 0. Since any path with backward movements cannot
be optimal, we can choose parametrization so that
x = x0 + ct
y=0 . (23)
z = const
N.N. Petrov [19–21] and R. Montgomery [18] proved independently that if this
path is short enough it is a solution to the minimization problem for some length
functional.
Example 2. Let the 2-dimensional distribution in R3 be defined by ω =
f (x)g(y)dx − xdy + dz, where f , g are some sufficiently smooth functions. The
tensions tensor is
f (x)g 0 (y) + 1
0
F = . (24)
−f (x)g 0 (y) − 1 0
If the equation f (x)g 0 (y) + 1 = 0 has a continuous solution for all y ∈ [y0 , y1 ]
with some y0 , y1 , then this is an abnormal geodesic. Since x = f −1 (−1/g 0 (y)),
we assume that g 0 : [y0 , y1 ] → R is a diffeomorphism to its image and there
is the inverse function f −1 continuous in the domain of −1/g 0 . Let us choose
g(y) = y 2 /2 and f (x) = x, then x = −1/y. If y = y0 + ct, then x = −1/(y0 + ct)
where c = const 6= 0 and t 6= −y0 /c. The coordinate z = z(t) is defined by the
horizontality condition dz = xdy − 21 xy 2 dx = − y1 dy + y2 d(− y1 ) = − y1 dy + 2y
1
dy =
1 1
− 2y dy. Hence, z(t) = z0 − 2 ln |y0 + ct|. We proved that this distribution admits
the abnormal geodesics
x = − y0 1+ t
y = y0 + t , t 6= −y0 , y0 , z0 ∈ R. (25)
z = z − 1 ln |y + t|
0 2 0
It is an open question whether some of these abnormal geodesics are solutions
of the minimization problem for some length functional on the distribution.
Further we consider regular geodesics only.
222 V. R. Krym
6 The Jacobi Equation
Let us assume that both the distribution and the metric tensor of the distri-
bution are independent of vertical coordinates (the cyclicity condition). Hence
the Lagrange multipliers are time-independent (constant). The Jacobi equation
for this class of distributions can be written in geometric covariant form. The
distribution is assumed to be totally nonholonomic. The Hilbert condition [2]
for geodesics is always fulfilled in this theory (all geodesics are non-singular).
Consider the minimization problem for the energy functional J(γ) =
1
RT 0 0
2 t0
hγ , γ i dt for horizontal paths with fixed endpoints and time. This is the
Lagrange problem. We shall omit the subscript α and the summation sign in
n n
lα ω α and lα F α .
P P
sums involving Lagrange multipliers such as
α=m+1 α=m+1
By definition, a two-parameter variation σ of a geodesic γ is a two-parameter
family of curves σ(t, µ, τ ) such that σ(t, 0, 0) ≡ γ(t) [2]. A geodesic γ and its
variation σ are assumed to be C 2 -smooth. In addition, we assume that the
3 3
third derivatives ∂ σ and ∂ σ do exist for all admissible t, µ, τ and are
∂µ∂t∂τ ∂t∂µ∂τ
continuous along γ. In the paper [11], it was shown that the second variation of
2 T
the energy functional for a distribution is as follows: ∂ J τ =0 = γ 0 , DZ +
∂µ∂τ ∂µ t0
µ=0
∂(ω(Z)) T
l + I(Y, Z), where Y = ∂σ and Z = ∂σ . The functional
∂µ t0 ∂µ ∂τ
Z TD
DY DZ E
I(Y, Z) = , − R(Y, γ 0 )γ 0 , Z dt−
t0 dt dt
Z T
DY
− l (∇Y F )(γ 0 , Z) + F , Z dt (26)
t0 dt
is called the index form of a geodesic γ with Lagrange multipliers l. The vector
fields Y (·, 0, 0) and Z(·, 0, 0) along γ will be denoted by the same letters Y , Z.
If one of the fields Y or Z is vertical, then I(Y, Z) = 0.
The metric tensor of a distribution is positively definite in sub-Riemannian
geometry. Therefore the functional I(Y, Y ) > 0 for variations of geodesics which
are sufficiently short. It is one of the necessary conditions of optimality. The
optimality may be lost if I(Y, Y ) = 0 and it will be lost if I(Y, Y ) < 0. To find
critical variations we should consider the minimization problem for the functional
If the distribution is integrable in some sence, i.e. the sequence of commutators (the
flag of the distribution) does not span the whole tangent bundle of the manifold,
then some vertical coordinates of Jacobi fields are dependend and the fundamental
matrix of the Jacobi equation is degenerated for all points.
The Schouten Curvature for a Nonholonomic Distribution 223
I(Y, Y ) with the constrains Φα (Y 0 , Y ) = 0 (1). Hence we should minimize
Z TD
1 DY DY E
Iλ (Y ) = , − R(Y, γ 0 )γ 0 , Y dt−
2
t0 dt dt
Z T Z T X n n
1 0 DY dY k X ∂ωk 0k j
− l (∇Y F )(γ , Y )+F , Y dt+ λ ωk + γ Y dt.
2 t0 dt t0 dt
k=1
∂xj j,k=1
Consider the variation Y 7→ Y + δY and collect linear for δY terms. We get
Z TD
DY DδY E
δIλ (Y ) = , − R(Y, γ 0 )γ 0 , δY dt−
t0 dt dt
Z T
1 DδY DY
l (∇δY F )(γ 0 , Y ) + (∇Y F )(γ 0 , δY ) + F
− ,Y + F , δY dt+
2 t0 dt dt
Z T X n k n
dδY X ∂ωk 0k j
+ λ ωk + γ δY dt. (27)
t0 k=1
dt
j,k=1
∂xj
Assuming that the vector field Y is C 1 -smooth and that the derivative DY
dt is
absolutely continuous, we have
T
Z T D
DY D DY E
δIλ (Y ) = , δY − , δY + R(Y, γ 0 )γ 0 , δY dt−
dt t0 t0 dt dt
T
Z T
l 1
l (∇δY F )(γ 0 , Y ) + (∇Y F )(γ 0 , δY ) − (∇γ 0 F ) δY, Y dt−
− F δY, Y −
2 t0 2 t0
Z T T
Z T
1 DY DY
λF δY, γ 0 dt. (28)
− l F δY, −F , δY dt + λω(δY ) +
2 t0 dt dt t0 t0
Since F = dω, we get (∇δY F )(γ 0 , Y ) + (∇Y F )(γ 0 , δY ) − (∇γ 0 F ) δY, Y =
(∇δY F )(γ 0 , Y ) + (∇γ 0 F ) Y, δY − (∇Y F )(δY, γ 0 ) = −2(∇Y F )(δY, γ 0 ).
Definition 1. A pair (Ỹ , λ), where Ỹ is a vector field along a geodesic γ with
Lagrange multipliers l, will be called a Jacobi field if Ỹ satisfies the variations
equations (1) and its horizontal projection Y = pr Ỹ satisfies the nonholonomic
Jacobi equation
D DY DY
+ R(Y, γ 0 )γ 0 + lF̂ + l(∇Y F̂ )(γ 0 ) + λF̂ (γ 0 ) = 0. (29)
dt dt dt
The F̂ operator is the tensions tensor F with the second index raised by the
inverse metric tensor of the distribution. We assume that both the distribution
and the metric tensor of the distribution are independent of vertical coordinates
(the cyclicity condition). Hence the Lagrange multipliers (both lα and λα ) are
time-independent. The equations (1), (29) together with λ0 = 0 are a system of
linear homogeneous differential equations with the variables (Ỹ , λ). The set of
224 V. R. Krym
solutions of this system is a linear space. Therefore there are two types of Jacobi
fields. For the first type λ ≡ 0 (zero vector). For the second type of Jacobi fields
λ 6= 0.
A horizontal vector field Y along a geodesic γ with Lagrange multipliers l
will be called a horizontal Jacobi field iff it satisfies (29) with some multipliers
λ.
Definition 2. Points t1 , t2 ∈ [t0 , T ], t1 6= t2 , are said to be conjugated along
a horizontal geodesic γ if there exists a nontrivial Jacobi field Y (with some λ)
along γ which vanishes at these points: Y (t1 ) = 0 and Y (t2 ) = 0.
A smooth variation σ(·, ·) : [t0 , T ] × (−, ) → N is a geodesic variation iff all
its longitudinal lines are geodesics.
Lemma 1. If σ(·, ·) : [t0 , T ] × (−, ) → N is a geodesic variation then the
horizontal projection of the field ∂σ is a horizontal Jacobi field along each lon-
∂τ
gitudinal line of this variation.
Proof. Let X = ∂σ , then due to the equations of geodesics DX ∂t + l(τ )F̂ X = 0. If
∂t
D DX D DX ∂σ
a field Y is horizontal then R(Y, X)X = ∂τ ∂t − ∂t ∂τ [11]. Let Y = pr ∂τ ,
2
D
therefore we can continue this equation R(Y, X)X = ∂τ (−l(τ )F̂ X) − D∂ 2Yt =
D2 Y
−lτ0 F̂ X − l(τ )(∇Y F̂ )X − l(τ )F̂ DY 0
∂t − ∂ 2 t . Assigning λ = lτ we get (29).
Theorem 3. Let γ be a geodesic with the origin x0 = γ(t0 ) and endpoint x1 =
γ(T ). The point x1 = explx0 (u) is conjugated with the point x0 along γ iff the
rank of the differential d(u,l) expx0 is not its maximum, i.e. iff (u, l) is a critical
point of the mapping (u, l) 7→ explx0 (u).
Proof. The Jacobi equation which appears in the Bliss accessory problem is the
result of the linearisation of geodesics equations
γ0 = X
DX
+ lF̂ X = 0 . (30)
dt
ω(X) = 0
The fundamental matrix in the Bliss problem matches the matrix of the dif-
ferential d(u,l) expx0 , because this differential satisfies the linearised geodesics
equations as the result of differentiation of a solution of a differential equation
by initial conditions [17, p. 289], [7]. Since we consider distributions with the
cyclicity condition the index form I(Y, Z) depends on the horizontal projections
of fields Y , Z only. Hence the horizontal projection of a Jacobi field in the
Bliss problem satisfies (29). Therefore we can consider a (horizontal) solution
of the equation (29) and find its vertical components by means of the varia-
tions equation (1). The fundamental matrix of the obtained solution matches
the fundamental matrix of the Bliss problem and therefore the differential ma-
trix d(u,l) expx0 .
In the next part of this paper we consider Jacobi fields of the first type.
The Jacobi fields of first type (λ ≡ 0).
The Schouten Curvature for a Nonholonomic Distribution 225
Lemma 2. If Y, Z are horizontal Jacobi fields along a geodesic γ with Lagrange
multipliers l, then f = hY, Z 0 i − hY 0 , Zi − lF (Y, Z) is a constant function.
Proof. f 0 = hY 0 , Z 0 i + hY, Z 00 i − hY 00 , Zi − hY 0 , Z 0 i − l(∇γ 0 F )(Y, Z) − lF (Y 0 , Z) −
lF (Y, Z 0 ) = −hY, R(Z, γ 0 )γ 0 + l(∇Z F̂ )γ 0 + lF̂ Z 0 i + hR(Y, γ 0 )γ 0 + l(∇Y F̂ )γ 0 +
lF̂ Y 0 , Zi − l(∇γ 0 F )(Y, Z) − lF (Y 0 , Z) − lF (Y, Z 0 ) = −l(∇Z F )(γ 0 , Y ) −
l(∇Y F )(Z, γ 0 ) − l(∇γ 0 F )(Y, Z) = −ldF (γ 0 , Y, Z) = 0.
Hence the Jacobi fields preserve the definite antisymmetric form. Note that
the vector field X = γ 0 is a horizontal Jacobi field. Indeed due to the geodesics
equations X 0 = −lF̂ X, therefore X 00 = −l(∇X F̂ )X − lF̂ X 0 , and this is the
equation (29) (the curvature contribution R(X, X)X = 0).
Applying Lemma 2 to an arbitrary horizontal Jacobi field Y and γ 0 we obtain
hY, γ 00 i − hY 0 , γ 0 i − lF (Y, γ 0 ) = C. Since γ is a horizontal geodesic, γ 00 = −lF̂ γ 0
and hY 0 , γ 0 i = const.
Lemma 3. A geodesic γ : [t0 , T ] → N may contain only a finite number of
points which are conjugated to t0 .
The proof is similar to that given in [3, p. 149].
If a (regular) geodesic γ : [t0 , T ] → N is a solution to the minimization
problem for the functional J, then the functional Iλ (Y ) is non-negative for any
vector field Y along γ satisfying (1).
Theorem 4. Let γ : [t0 , T ] → N be a geodesic and the semi-interval (t0 , T ] does
not contain points conjugated with t0 . Then the functional Iλ (Y ) is positively
definite for all vector fields Y along γ satisfying (1) wich are zero at the endpoints
of γ.
Theorem 5. Let the cyclicity condition be satisfied for a distribution. Suppose
that the metric tensor of a distribution is positive definite, a (regular) geodesic γ
connects two given points x0 and x1 , and there are no points on the semi-interval
(t0 , T ] that are conjugated to t0 . Then, on the path γ, the energy functional
attains its weak local minimum in the problem with fixed endpoints.
The proof is given in [12].
7 Conclusion
Hence we propose the Jacobi equation for horizontal geodesics on a distribution
in sub-Riemannian geometry which involves the curvature tensor of a distribu-
tion and its tensions tensor. The classical variational theory treats this equation
as a sum of derivatives of the given functional. We established the geomet-
ric sense of these sums in terms of well-defined geometric objects. The Jacobi
equation (29) is applicable to regular geodesics. Since there are also abnormal
geodesics in the sub-Riemannian geometry, we discussed two examples of such
geodesics. There is one more way where geodesics may loss its optimality: it is
the case when two given points can be connected by more than one geodesic.
We could not discuss this case here yet it should be always noted.
226 V. R. Krym
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