The paper shows that if the distribution is de ned on a manifold with the special smooth structure and does not depend on the vertical 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 subRiemannian geometry. Conjugate points are de ned by the solutions of the Jacobi equation. If a geodesic passed a point conjugated with its beginning then this geodesic ceases to be optimal.
Distribution on a smooth manifold N is a family of subspaces A(x) TxN , x 2 N . General theory of the variational calculus with nonholonomic restrictions '(t; x; x_ ) = 0 was published by G.A. Bliss [2]. If the restrictions are linear for velocities, !x(x_ ) = 0, where ! is a 1-form, we get distributions [25, 26].
The Romanian mathematician Vranceanu rst introduced the term of the nonholonomic structure on a Riemannian manifold in 1928 [27]. The Dutch mathematician Schouten de ned the connection and appropriate curvature tensor for horizontal vector elds on a distribution in [23, 22]. The soviet geometrician 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 di erent from the Riemannian curvature. In this paper we prove that if the distribution is de ned 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 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 de ned by means of the solutions of the Jacobi equation. In this paper we discuss several examples of conjugate points in sub-Riemannian geometry. 2
Let : [t0; T ] ! N be a C1-smooth horizontal path, !( 0) = 0, where ! is a 1form. Horizontal variation of is a 1-parametric family of maps ( ; ) : [t1; t2] ! N , j j < ", if there are continuous vector elds X = @@t , Y = @@ , the central line is just (t; 0) = (t) and the eld X is horizontal, X(t; ) 2 A( (t; )) and n X ! k=1
j;k=1
= m+1; : : :; n:
(
The Lagrange Problem Let us consider the 2-dimensional distribution A on R3 de ned by the 1-form ! = x2dx1 + dx3. The energy functional is J (x( )) = 12 R0T (x_ 1)2 + (x_ 2)2 dt, the metric tensor for this distribution is the identity matrix. The Lagrangian of the variational problem is L = 12 (x_ 1)2 + (x_ 2)2 + l!( 0). The horizontal geodesics starting from the origin in R3 for l 6= 0 are 8 x1(t) = 1l > >>> x2(t) = 1l v1 < > > > > : v2 + v2 cos lt + v1 sin lt ;
v1 cos lt + v2 sin lt ; +2v1v2 cos 2lt + (v12
v22) sin 2lt :
x3(t) = 41l2 2lt(v12 + v22) + 2v1v2
4v1v2 cos lt
4v12 sin lt+
(
Z T Xn
t0 i;j=1 =0 t0 = L(t; x; x_ ) 0
+2
T
t0
where (t0) = ddt0 , (T ) = ddT , 0 (t0) = dd2t20 , 0 (T ) = dd2T2 , = @@ at = 0.
This functional should be minimized for variables satisfying the equations of
variations along [2]. Since the distribution A is de ned by the 1-form ! =
x2dx1 + dx3 in our example, the constrains (
_3 + 1l v1 >>: _ = 0: v1 sin lt + v2 cos lt = 0 v1 cos lt v2 sin lt = 0 v1 cos lt + v2 sin lt _1 + v1 cos lt v2 sin lt 2 = 0 This is the system of linear homogeneous di erential equations for the variables ; . To nd points conjugate with the starting point t0 = 0 one should nd the solutions of these equations which are zero (0) = 0 at the initial point but 0(0) 6= 0. The solutions have the form
( 1; 2; 3)T = P (a1; a2; )T ;
where a1, a2, are constants and P is the following part of the fundamental
matrix of the system (
BB 1 clos lt P = BBB l
B 2v1lt 4v1 sin lt+ B +v1 sin 2lt 2v2 cos lt+ @ +v2 cos 2lt+v2 2l2 cos lt 1
l sin lt
l 2v2lt v2 sin 2lt+ +v1 cos 2lt+v1 2v1 cos lt 2l2 (v1lt (v1lt v2) cos lt (v1 + v2lt) sin lt + v2 1
l2 v2) sin lt + (v1 + v2lt) cos lt v1 CC
l2 CC :
(v12+v22)lt 4v12 sin lt+(v12 v22+2v1v2lt) sin 2lt+ CC
+2(v1lt +2(v22v)1vv12c+o(svl22t 2vv121)lvt2)lctossin2lltt+2v1v2+ AC
2l3
If t1 is a point conjugated with the initial point then there is the solution of (
sin(
lt
2
lt
2
lt
2
)(v12 + v22)=l4:
Since det P (t1) = 0 we get two series of conjugate points. For the conjugate
points of rst series sin l2t = 0 and tk = 2 k=l, k 2 Z. The rst conjugate point
of this series is t1 = 2 =l (Fig. 1). The appropriate Jacobi eld is de ned by (
The equations (
l t cos lt)v1 ( sin lt l t cos lt)v2 + ( l t sin lt
cos lt + )v1 b1l sin lt
b2l cos lt + b2l =l2; = ( sin 2lt l t cos 2lt
l t)v22+ =0 (( 2l t sin 2lt 2 cos 2lt + 2l t sin lt + 4 cos lt 2 )v1 b2l sin 2lt + b1l cos 2lt
2b1l cos lt + 2b2l2t + b1l)v2+ ( sin 2lt + l t cos 2lt + 4 sin lt 2l t cos lt l t)v12+ (b1l sin 2lt + b2l cos 2lt 4b1l sin lt 2b2l cos lt + 2b1l2t + b2l)v1 =(2l3): some
and b. For the conjugate points of rst series this is
The rank of exponential geodesic map is not its maximum rank at a conjugate
=0
(t1) = 0 for
8 (2 v1)=l2 = 0
< (2 v2)=l2 = 0
: (2 ( (v12 + v2 )
2
l(b2v2 + b1v1)))=l3 = 0:
(
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 su ciently
small domain U N can be chosen so as to maximize the projection of the
distribution A on the rst m coordinates. Then the basis of the distribution A
can be chosen as
n
X
=m+1
k = 1; : : :; m;
(
For any point x 2 N the quadratic form h ; ix is de ned on A(x). The metric
tensor in the basis (
gij (x) = hei; ej ix:
To de ne covariant di erentiation r on the distribution, we must introduce a symmetric Riemannian connection. The property of being Riemannian is de ned in a standard way: while the symmetry condition must be modi ed as
XhY; Zi = hrX Y; Zi + hY; rX Zi; rX Y
rY X = pr([X; Y ]); m where pr = P ek dxk is the (horizontal) projection on the distribution. To 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: @@yxk = 0, k = 1; : : :; m, = m+1; : : :; n, and @@xy ; =m+1;:::;n is the identity matrix [13, 14, 16]. The coordinates x will be called verticals. The di erentials of the transfer maps hU hV 1 for all charts hU : U ! Rn, U; V N , admissible in this smooth structure are the block matrices
0 1
(
Since the connection r is Riemannian and pr-symmetric (
hZ; pr[X; Y ]i): (
For any point x 2 N and any three vectors u; v; w 2 A(x) the curvature map
of distribution A at point x is de ned by the Schouten tensor [23, 22, 6]
R(u; v)w = ru~rv~w~
rv~ru~w~
rpr[u~;v~]w~
pr (1
pr)[u~; v~]; w~ ;
(
Let the distribution A on a manifold N n be de ned by the di erential 1-forms m ! = P As dxs + dx , = m+1; : : :; n. The geodesics equations for a distribus=1 tion with the cyclicity condition are [10] where dDt { covariant derivative, (a0; l) { the Lagrange multipliers, which cannot be altogether zero. A geodesic is called normal (or regular), i there is the only one set of multipliers (a0; l) with a0 = 1 for . Operator F^ is the tensions tensor Fij = @jAi @iAj with the second index raised by the inverse metric tensor of the distribution.
Example 1. Let the 2-dimensional distribution in R3 be de ned by ! = 0 2y y2dx+dz. The tensions tensor F = 2y 0 . Non-trivial abnormal geodesics appear at y = 0. It can be any absolutely continuous function x = x(t) with z = const due to !( 0) = 0. Since any path with backward movements cannot be optimal, we can choose parametrization so that 8 x = x0 + ct <
y = 0 : 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 de ned by ! = f (x)g(y)dx xdy + dz, where f , g are some su ciently smooth functions. The tensions tensor is
F =
0
f (x)g0(y) 1
f (x)g0(y) + 1
0
:
21y dy. Hence, z(t) = z0
the abnormal geodesics
If the equation f (x)g0(y) + 1 = 0 has a continuous solution for all y 2 [y0; y1]
with some y0, y1, then this is an abnormal geodesic. Since x = f 1( 1=g0(y)),
we assume that g0 : [y0; y1] ! R is a di eomorphism to its image and there
is the inverse function f 1 continuous in the domain of 1=g0. Let us choose
g(y) = y2=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 de ned by the
horizontality condition dz = xdy 12 xy2dx = y1 dy + y2 d( y1 ) = y1 dy + 21y dy =
12 ln jy0 + ctj. We proved that this distribution admits
(
y = y0 + t
>: z = z0 21 ln jy0 + tj
; t 6=
y0; y0; z0 2 R:
(
Further we consider regular geodesics only.
l
I(Y; Z) = Let us assume that both the distribution and the metric tensor of the distribution 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 ful lled in this theory (all geodesics are non-singular).
Consider the minimization problem for the energy functional J ( ) = 12 RtT0 h 0; 0i dt for horizontal paths with xed endpoints and time. This is the Lagrange problem. We shall omit the subscript and the summation sign in n n sums involving Lagrange multipliers such as P l ! and P l F . =m+1 =m+1
By de nition, a two-parameter variation of a geodesic is a two-parameter family of curves (t; ; ) such that (t; 0; 0) (t) [2]. A geodesic and its variation are assumed to be C2-smooth. In addition, we assume that the @3 @3 third derivatives @ @t@ and @t@ @ do exist for all admissible t, , and are continuous along . In the paper [11], it was shown that the second variation of T
+ t0 =0 = =0 Z T D DY t0 dt ;
DZ E dt
R(Y; 0) 0; Z
dt Z T t0 l (rY F )( 0; Z) + F ; Z
dt (
The metric tensor of a distribution is positively de nite in sub-Riemannian geometry. Therefore the functional I(Y; Y ) > 0 for variations of geodesics which are su ciently 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 nd 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 ag of the distribution) does not span the whole tangent bundle of the manifold, then some vertical coordinates of Jacobi elds are dependend and the fundamental matrix of the Jacobi equation is degenerated for all points.
Consider the variation Y 7! Y + Y and collect linear for Y terms. We get Z T D DY t0 dt ;
D Y E dt
R(Y; 0) 0; Y l (r Y F )( 0; Y ) + (rY F )( 0; Y ) + F ; Y + F ; Y
dt+ dt D Y dt Assuming that the vector eld Y is C1-smooth and that the derivative DdYt is absolutely continuous, we have
I (Y ) =
1 Z T
De nition 1. A pair (Y~ ; ), where Y~ is a vector eld along a geodesic with
Lagrange multipliers l, will be called a Jacobi eld if Y~ satis es the variations
equations (
D DY
dt dt
The F^ 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 l and ) are
time-independent. The equations (
A horizontal vector eld Y along a geodesic with Lagrange multipliers l will be called a horizontal Jacobi eld i it satis es (29) with some multipliers .
De nition 2. Points t1; t2 2 [t0; T ], t1 6= t2, are said to be conjugated along a horizontal geodesic if there exists a nontrivial Jacobi eld Y (with some ) along which vanishes at these points: Y (t1) = 0 and Y (t2) = 0.
A smooth variation ( ; ) : [t0; T ] ( its longitudinal lines are geodesics.
; ) ! N is a geodesic variation i all ; ) ! N is a geodesic variation then the is a horizontal Jacobi eld along each lonl( )(rY F^)X , Theorem 3. Let be a geodesic with the origin x0 = (t0) and endpoint x1 = (T ). The point x1 = explx0 (u) is conjugated with the point x0 along i the rank of the di erential d(u;l) expx0 is not its maximum, i.e. i (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 8
0 = X
< DdXt + lF^X = 0 :
: !(X) = 0
(30)
The fundamental matrix in the Bliss problem matches the matrix of the
differential d(u;l) expx0 , because this di erential satis es the linearised geodesics
equations as the result of di erentiation of a solution of a di erential 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 elds Y , Z only. Hence the horizontal projection of a Jacobi eld in the
Bliss problem satis es (29). Therefore we can consider a (horizontal) solution
of the equation (29) and nd its vertical components by means of the
variations equation (
In the next part of this paper we consider Jacobi elds of the rst type. The Jacobi elds of rst type ( 0). Lemma 2. If Y; Z are horizontal Jacobi elds along a geodesic with Lagrange multipliers l, then f = hY; Z0i hY 0; Zi lF (Y; Z) is a constant function. Proof. f 0 = hY 0; Z0i + hY; Z00i hY 00; Zi hY 0; Z0i l(r 0 F )(Y; Z) lF (Y 0; Z) lF (Y; Z0) = hY; R(Z; 0) 0 + l(rZ F^) 0 + lF^Z0i + hR(Y; 0) 0 + l(rY F^) 0 + lF^Y 0; Zi l(r 0 F )(Y; Z) lF (Y 0; Z) lF (Y; Z0) = l(rZ F )( 0; Y ) l(rY F )(Z; 0) l(r 0 F )(Y; Z) = ldF ( 0; Y; Z) = 0.
Hence the Jacobi elds preserve the de nite antisymmetric form. Note that the vector eld X = 0 is a horizontal Jacobi eld. Indeed due to the geodesics equations X0 = lF^X, therefore X00 = l(rX F^)X lF^X0, and this is the equation (29) (the curvature contribution R(X; X)X = 0).
Applying Lemma 2 to an arbitrary horizontal Jacobi eld Y and 0 we obtain hY; 00i hY 0; 0i lF (Y; 0) = C. Since is a horizontal geodesic, 00 = lF^ 0 and hY 0; 0i = const.
Lemma 3. A geodesic : [t0; T ] ! N may contain only a points which are conjugated to t0. nite number of The proof is similar to that given in [3, p. 149].
If a (regular) geodesic : [t0; T ] ! N is a solution to the minimization
problem for the functional J , then the functional I (Y ) is non-negative for any
vector eld Y along satisfying (
Theorem 4. Let : [t0; T ] ! N be a geodesic and the semi-interval (t0; T ] does
not contain points conjugated with t0. Then the functional I (Y ) is positively
de nite for all vector elds Y along satisfying (
Theorem 5. Let the cyclicity condition be satis ed for a distribution. Suppose that the metric tensor of a distribution is positive de nite, a (regular) geodesic 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 , the energy functional attains its weak local minimum in the problem with xed endpoints.
7
Hence we propose the Jacobi equation for horizontal geodesics on a distribution in sub-Riemannian geometry which involves the curvature tensor of a distribution and its tensions tensor. The classical variational theory treats this equation as a sum of derivatives of the given functional. We established the geometric sense of these sums in terms of well-de ned 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.