Table Of ContentSEMI - RIEMANNIAN GEOMETRY WITH NONHOLONOMIC
CONSTRAINTS
ANNAKOROLKO,IRINAMARKINA
9
0
Abstract. In thepresentarticle thegeometry of semi-Riemannian manifolds with nonholo-
0
nomic constraints is studied. These manifolds can be considered as analogues to the sub-
2
Riemannianmanifolds, wherethepositively definitemetricissubstitutedbyanondegenerate
n
metric. To study properties of the exponential map the Christoffel symbols and other differ-
a
ential operators were introduced. We study solutions of the Hamiltonian system and their
J
projections into the underlying manifold. The explicit formulae were found for a specific
1
example of a semi-Riemannian manifold with nonholonomic constraints.
1
]
G
D 1. Introduction
.
h
Sub-Riemannianmanifoldsandthegeometry introducedbybracketgenerating distributions
t
a of smoothly varying m-dimensional planes is widely studied interesting subject, which has
m
applicationsincontroltheory,quantumphysics,C-Rgeometry, thetheoryofprincipalbundles,
[ and other areas. The main difference of the sub-Riemannian manifold from a Riemannian one
1 is the presence of a smooth subbundle of the tangent bundle, generating the entire tangent
v bundle by means of the commutators of vector fields. The subbundle, which is often called
7
horizontal, is equipped with a positively definite metric that leads to the triple: manifold,
7
4 horizontal subbundle, and Riemannian metric on the horizontal subbundle, which is called
1 a sub-Riemannian manifold. The foundation of the sub-Riemannian geometry can be found
.
1 in [12, 13, 14, 17]. The following question can be asked. What kind of geometrical features
0 will have the mentioned triplet if we change the positively definite metric on the subbundle to
9
an indefinite nondegenerate metric? We use the term semi-Riemannian to emphasis that the
0
: considered metric is nondegenerate in contrast to the positively definite metric, that referred
v
as Riemannian. As it is known to the authors the present work is the first attempt to study
i
X systematically the geometry of semi-Riemannian manifolds with nonholonomic constraints,
r that we called the sub-semi-Riemannian manifolds or shortly ss-manifolds. In the present
a
paper we study the exponential map and solutions to the Hamiltonian system that has no
established terminology in the literature and can be called geodesics or extremals, see, for
instance [12,17]. Thess-manifolds have their own peculiarity thatdistinguishes them from the
sub-Riemannianandsemi-Riemannianmanifolds. Thesimplestexampleofasemi-Riemannian
manifold with nonholonomic constraints is the Heisenberg group equipped with the Lorentzian
metric and it has been considered in [6, 7, 11]. It was shown in [11] that in contrast with the
Heisenberg group with positively definite metric the Lorentzian type of the Heisenberg group
possesses the uniqueness of extremals both of timelike or spacelike type. The structure of the
2000 Mathematics Subject Classification. 53C50, 53B30 53C17.
Key words and phrases. Semi-Riemannian manifolds, nondegenerate metric, exponential map, Christoffel
symbol, extremals, quaternions.
The authors are partially supported by the grant of the Norwegian Research Council # 177355/V30, by
thegrantoftheEuropean ScienceFoundationNetworkingProgrammeHCAA,andbytheNordForskResearch
Network Programme # 080151.
1
2 ANNAKOROLKO,IRINAMARKINA
article is the following. Section 2 is devoted to main definitions. The collection of technical
lemmas concerning Christoffel symbols is proved in Section 3. In Section 4 the extremals and
exponential map are introduced, the extendability of extremals and Gauss lemma are shown.
Somepropertiesofthelengtharealsostudied. Section5isdevotedtothedifferentialproperties
of the exponential map. It is shown that the exponential map possesses an analogue of “local
diffeomorphism”property, although is itnotadiffeomorphismat theorigin. ThelastSection 6
consists of the example of ss-manifold, where the explicit formulae of extremals are found.
2. Main definitions
Let M be a connected n-dimensional, n ≥ 3, C∞-manifold. Let T and T∗ denote the
x x
tangent and cotangent spaces at a point x∈ M, and hY,ξi the pairing between them, Y ∈ T ,
x
ξ ∈ T∗. The tangent and cotangent bundles are denoted by T and T∗ respectively. Fix an
x
integer m, 1 < m < n. Let S be a fixed subbundle of the tangent bundle T, S = S , S
x x
x∈SM
be a fiber over x, of the rank m. A subbundle S will be called bracket generating or complete
nonholonomic, if the vector fields which are sections of S, together with all brackets span T
x
at each x ∈ M. In this case any two points in M can be connected by a piecewise smooth
curve γ(s) such that the tangent vector γ˙(s) belongs to S at each point γ(s) where the
γ(s)
tangent vector exists. The bracket generating subbundle S is called the horizontal bundle
or horizontal distribution and a curve γ(s) satisfying γ˙(s) ∈ S is called the horizontal
γ(s)
curve. A result of Chow [4], see also [16], guarantees the connectivity of M by a horizontal
curve. The necessary and sufficient condition on connectivity by curves tangent to a given
distribution of a smooth manifold can be found in [18]. We notice that the connectivity of
a manifold by horizontal curves tangent to a given distribution depends only on properties
of the distribution and not on any metric defined on it or on the tangent bundle. If Y ∈ S,
let S +[Y,S] denote the subbundle of T spanned by S and all the vector fields [Y,X], where
X varies over sections S. A fiber at a point x ∈ M is written as S +[Y(x),S ] ∈ T with
x x x
Y(x) ∈ S . Similarly we define bracket(k,Y) inductively by bracket(2,Y) = S +[Y,S] and
x
bracket(k,Y)= S+[bracket(k−1,Y),S]. More generally we set bracket(2,S) = S+[S,S] and
bracket(k,S) = S +[bracket(k −1,S),S]. A restriction of a bundle to x ∈ M is denoted by
writing the subscript x, for example: bracket(k,Y(x)) = S +[bracket(k−1,Y(x)),S ] ∈ T .
x x x
WewillsaythatavectorfieldY ∈S isak-stepgeneratorifbracket(k,Y(x)) = T forallx∈ M.
x
Similarly, S will be said to be k-step bracket generating distribution if bracket(k,S )= T for
x x
everyx. FromnowonweworkwithadistributionS possessingthebracketgeneratingproperty.
By analogy with the sub-Riemannian metric we give the following definition.
Definition 2.1. Let M be a smooth manifold, S be a bracket generating subbundle of the
tangent bundle T. A sub-semi-Riemannian metric Q on S is a smoothly varying in x non-
degenerate quadratic form Q on S . We abbreviate the long and tedious name of sub-semi-
x x
Riemannian metric by the term ss-metric. We call the pair (S,Q) the sub-semi-Riemannian
(ss-Riemannian) structure on M.
Weremindthattheindexν ofametricisthemaximaldimensionofthespaceV ⊂ S ,where
x x
theformQ isnegatively definite. Ifν = 1thenwecall thess-metricthesub-Lorentzian metric
x
following the tradition in semi-Riemannian geometry. The ss-metric with the index ν = 0 is
just a sub-Riemannian metric. Given Q , we may define a linear mapping g : T∗ → T as
x x x x
follows: forgiven ξ ∈ T∗,thelinearmappingW → hW,ξi, W ∈ S canberepresenteduniquely
x x
as W → Q (W,X) for some X ∈ S , then X is chosen to be g ξ. The map g is called a
x x x x
cometric and completely defined by the two following relations:
SEMI-RIEMANNIAN GEOMETRY WITH CONSTRAINT 3
(i) image of T∗ under g is S ,
x x x
(ii) g and Q are related by the identity
x x
(2.1) Q (W,g ξ) = hW,ξi for all W ∈ S .
x x x
Lemma 2.2. If Q is symmetric, nondegenerate and has index ν, then g is symmetric,
x x
nondegenerate and has index ν.
Proof. We understand the action of the cometric g on T∗×T∗ → R (omitting x) as following:
g(ξ,ψ) = hgξ,ψi for any two covectors ξ and ψ from T∗.
Thus by definition of the cometric g we have g(ψ,ξ) = hgψ,ξi = Q(gψ,gξ), which equals to
Q(gξ,gψ) = hgξ,ψi = g(ξ,ψ) by the symmetry of the ss-metric Q.
Now, having the nondegeneracy of Q we prove the nondegeneracy of g, that is if g(ξ,ψ) = 0
for any ψ ∈ T∗ then ξ ≡ 0. First of all, we notice that the pairing hY,ξi is not degenerate
by (2.1). Then, taking arbitrary ψ ∈ T∗ and setting Y = gψ, we obtain
g(ψ,ξ) = hY,ξi = 0 for any Y ∈ S.
This implies that ξ ≡ 0 by the nondegeneracy of pairing and we conclude that g is nondegen-
erate cometric.
Let S⊥ denote the kernel of g , and S⊥ ⊆ T∗ be the subbundle with fibers S⊥. Then
x x x
g : T∗/S⊥ → S is bijection. The relation (2.1) shows that the index of Q and g coincides
x x x x x x
for any x∈ M and that g degenerates on S⊥. (cid:3)
x x
Conversely, given a symmetric nondegenerate linear operator g : T∗ → T with image S ,
x x x x
jk
thereisauniquenondegeneratequadraticformQ satisfying (2.1). Wewriteg ,j,k = 1,...n
x x
for the symmetric matrix defining the cometric g to emphasis that it is a tensor of covariant
x
jk
type and operates with covectors. The matrix g is never invertible.
x
Adifferential manifold M with achosen subbundleS of thetangent bundleandwithagiven
nondegeneratess-metricQonS willbecalled thesub-semi-Riemannian manifoldorshortlyss-
manifold. If the index ν of Q is 1, then we call the triplet (M,S,Q) a sub-Lorentzian manifold
and in the case of ν = 0 we get the sub-Riemannian manifold widely studied in [9, 12, 14, 17]
and numerous references therein.
We present a couple of examples of ss-manifolds.
Example 1.
Letusconsiderthefollowingexampleofsub-LorentzianmanifoldthatwecalltheHeisenberg
group with sub-Lorentzian metric. This example was considered first in [6, 7] and was also
studied in [11]. We remind that the Heisenberg group H1 is the space R3 furnished with the
non-commutative law of multiplication
1
(x,y,z)(x′,y′,z′) = x+x′,y+y′,z+z′+ (yx′−xy′) .
2
(cid:0) (cid:1)
This gives the R3 the structure of a non-abelian Lie group. The two-dimensional horizontal
bundle S is given as a span of left invariant vector fields
∂ 1 ∂ ∂ 1 ∂
X = + y , Y = − x ,
∂x 2 ∂z ∂y 2 ∂z
that can be found as the left action of the Lie group. There is only one nonvanishing com-
mutator [X,Y] = Z = ∂ . We suppose that the Lorentzian metric Q is defined on S by
∂z
setting
Q(X,X) = −1, Q(Y,Y)= 1, Q(X,Y) = 0.
4 ANNAKOROLKO,IRINAMARKINA
Thus the triple (R3,S,Q) is called the Heisenberg group with the Lorentzian metric, and to
differ it from the classical case H1 we use the notation H1. We say the classical case bearing
L
in mind the manifold (R3,S,d) with a positively definite metric d on S.
The quadratic nondegenerate symmetric form Q on S is of the form
−1 0
Q = {Q } = .
ab (cid:18) 0 1(cid:19)
TakethebasisofLiealgebraassociatedwiththeHeisenberggroup,consideredastheLiegroup,
(X,Y,Z) ∈ T. The dual basis of T∗ consists of the forms dx, dy, ω = dz− 1(xdy−ydx). We
2
wish to find the cometric g = gjk. Let gdx = a X+a Y. Making use of (2.1) for W replaced
1 2
by X and Y, we deducethat gdx = −X. In the same way we get gdy = Y. Thus,the equality
g(ζ,ξ) = Q(gζ,gξ) calculated for the basic forms implies the values g = gjk for j,k = 1,2.
The rest of the terms vanish because of g(dx,ω) =hgdx,ωi = 0, g(dy,ω) = hgdy,ωi = 0, and
gω = 0. Finally we get
−1 0 0
gjk = 0 1 0.
0 0 0
Example 2. Consider the example of ss-manifold related to the notion of Heisenberg-type
groups based on quaternions [2, 5, 10]. The manifold M is R7. The vector fields
∂ 1 ∂ ∂ ∂
X = + +x −x −x ,
1 2 4 3
∂x 2 (cid:18) ∂z ∂z ∂z (cid:19)
1 1 2 3
∂ 1 ∂ ∂ ∂
X = + −x −x +x ,
2 1 3 4
∂x 2 (cid:18) ∂z ∂z ∂z (cid:19)
2 1 2 3
∂ 1 ∂ ∂ ∂
X = + +x +x +x ,
3 4 2 1
∂x 2 (cid:18) ∂z ∂z ∂z (cid:19)
3 1 2 3
∂ 1 ∂ ∂ ∂
X = + −x +x −x ,
4 3 1 2
∂x 2 (cid:18) ∂z ∂z ∂z (cid:19)
4 1 2 3
form the basis of four-dimensional horizontal distribution S. These vector fields come from
the infinitesimal action of the noncommutative group law multiplication
1
L (x′,z′) = (x,z)◦(x′,z′)= x+x′,z+z′+ Im(x¯∗x′)
(x,z) 2
(cid:0) (cid:1)
for (x,z) and (x′,z′) from R4×R3. Here Im(x¯∗x′) is the imaginary part of the product x¯∗x′
of the conjugate quaternion x¯ to x by another quaternion x′. See the details in Section 6. The
distribution S is bracket generating due to the commutation relations
[X ,X ] = −Z , [X ,X ] = Z , [X ,X ]= Z ,
1 2 1 1 3 3 1 4 2
[X ,X ]= Z , [X ,X ] = −Z , [X ,X ] = −Z ,
2 3 2 2 4 3 3 4 1
where Z = ∂ , β = 1,2,3 form a basis of the complement to S in the tangent bundle.
β ∂zβ
We define the ss-metric Q on S by the matrix
−1 0 0 0
0 −1 0 0
Q = .
αβ 0 0 1 0
0 0 0 1
SEMI-RIEMANNIAN GEOMETRY WITH CONSTRAINT 5
Thess-metricQhasindex2. Thecorrespondingcometricgjk isobtainedlikeintheExample1,
has index 2, and assumes the following form
−1 0 0 0 0 0 0
0 −1 0 0 0 0 0
0 0 1 0 0 0 0
g = 0 0 0 1 0 0 0.
jk
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
More details about the manifold of the Example 2 the reader can find in Section 6.
We refer to [3] for an example of sub-Lorentzian manifold based on the Lie group different
from the nilpotent group.
3. Christoffel symbols
Recall that S⊥ denotes the kernel of g(x) and S⊥ ⊆ T∗, S⊥ = ∪ S⊥. The space S⊥ is
x x∈M x x
the annihilator of S in T∗. From now on, we use the summation convention of the differential
x x
geometry.
Lemma 3.1. Let v be a covector from S⊥. Then hv,Yi = 0 for all Y ∈ S if and only if
gjkv = 0, j = 1,...,n.
k
Proof. Let hv,Yi = 0 for any Y ∈ S. Pick up an arbitrary Y ∈ S, then there exists a form ω
such that Y = gω. Moreover, we can assume that ω ∈ T∗/S⊥. Indeed, if ω = ω +v , where
1 1
ω ∈ T∗/S⊥ and v ∈S⊥, then for any X ∈ S we get
1 1
Q(X,gω) = Q(X,gω )+Q(X,gv ) = Q(X,gω )+hv ,Xi = Q(X,gω ).
1 1 1 1 1
Thus
0 = hv,Yi =hv,gωi = Q(gω,gv) for all Y = gω ∈ S ⇒ gv = 0.
Here we used the symmetry and the nondegeneracy of Q.
Conversely, having gv = 0 we derive 0 = hv,gωi for any ω ∈ T∗/S⊥. Thus, hv,Yi = 0 for
any Y = gω. (cid:3)
Lemma 3.2. (a) If v is a section of the annihilator S⊥, then
∂v ∂gjk
gjk k = − v .
∂xp ∂xp k
(b) If x(t) is a curve in M and v(t) is such that v(t) is a section of S⊥ over x(t), then
∂gjk(x)
gjk(x)v˙ = − x˙pv
k ∂xp k
for all t (here the dot denotes the t-derivative).
(c) If v and w are sections of S⊥, then
∂gjk
v w = 0.
∂xp k j
6 ANNAKOROLKO,IRINAMARKINA
∂
Proof. To prove (a) one applies to the identity gjk(x)v (x) = 0 which defines the null-
∂xp k
bundle.
d
To prove (b) we take the derivative of the identity gjk(x(t))v (t) = 0.
k
dt
Finally, to prove (c) first we apply (a) to obtain
∂v ∂gjk
gjk k = − v .
∂xp ∂xp k
Then we multiply both sides by w ∈ S⊥ and, making use of the symmetry of g, we get
∂gjk ∂v ∂v
v w = −gjk kw =−gkjw k = 0
∂xp k j ∂xp j j∂xp
because of gkjw = 0. (cid:3)
j
The following question can arise: how the information about the bracket generating prop-
erties reflects in the properties of g? If X, Y ∈ S, then [X,Y] is an element of T/S. If X ∈S
x
then there exists ξ ∈ T∗ with X = g ξ, and similarly Y = g η. The covectors ξ and η are not
x x x
defined uniquely, as it was shown in the proof of Lemma 3.1. Thus they should be regarded
as elements of T∗/S⊥. The annihilator S⊥ contains all the necessary information concerning
x x
the commutators through the pairing h[X,Y],vi, when v varies over S⊥. Let us consider the
trilinear form h[gξ,gη],vi on (T∗/S⊥)×(T∗/S⊥)×S⊥.
x x x x x
Lemma 3.3. In local coordinates
∂grq ∂grp
(3.1) h[gξ,gη],vi = gjp −gjq ξ η v
(cid:18) ∂xj ∂xj (cid:19) p q r
for v varying over S⊥ for any x.
x
Proof. Let ξ and η denote any sections of T∗. Then Xr = grpξ and Yr = grqη are sections
p q
of S, and
∂ ∂
[X,Y]r = Xj Yr −Yj Xr
∂xj ∂xj
∂grq ∂grp ∂η ∂ξ
= gjpξ η −gjqη ξ +gjpξ grq q −gjqη grp p.
p ∂xj q q ∂xj p p ∂xj q ∂xj
Taking the inner product with v ∈ S⊥, we find that the last two terms are annihilated since
∂η ∂η ∂ξ ∂ξ
grq qv = gqrv q = 0 and grp pv = gprv p = 0. Thus we obtain (3.1). (cid:3)
∂xj r r∂xj ∂xj r r∂xj
We want do define the analogue of the Christoffel symbols but with the raised indexes and
see the relation between them and the trilinear form defined in Lemma 3.3. We write
1 ∂gpq ∂gkq ∂gkp
(3.2) Γkpq = gkj −gpj −gqj .
2(cid:18) ∂xj ∂xj ∂xj (cid:19)
For sections ξ ∈ T∗ and v ∈ S⊥ define Γ(ξ,v) ∈ T by Γk(ξ,v) = Γkpqξ v . In classical case
p q
of differential geometry the Christoffel symbols are used to express the covariant derivative in
local coordinates. Unlike to the classical covariant derivative, which associates for two vector
fields another vector field, the operator Γ, as we will see from the following lemma, associates
a vector field for a pair of covector fields and, moreover, the resulting vector field is horizontal.
Lemma 3.4. Γ(ξ,v) is a well-defined vector field; that is it is independent of the choice of
coordinates. Moreover, Γ(ξ,v) is a horizontal vector field and Γ(ξ+w,v) = Γ(ξ,v) for w ∈ S⊥,
so that Γ: (T∗/S⊥)×S⊥ → S.
SEMI-RIEMANNIAN GEOMETRY WITH CONSTRAINT 7
Proof. Let us prove that Γk(ξ,v) transforms as a tangent vector at each x: Γk(y) = Γd(x)∂yk,
∂xd
where y = ψ(x) and ψ is a local diffeomorphism determining a new coordinate system. By
e
g, ξ and v denote the expressions for g, ξ and v in the new coordinates. We have
(e3.3e) e ξ = ∂ψj(x)ξ , v = ∂ψj(x)v , gkj(y) = gpq(x)∂ψk(x)∂ψj(x).
k ∂xk j k ∂xk j ∂xp ∂xq
In the new coordinates e e e
∂gkq ∂yj ∂yp ∂ ∂yk ∂yq ∂xl
gjp = gab gcd ,
∂yj (cid:18) ∂xa ∂xb(cid:19) ∂xl (cid:18) ∂xc ∂xd(cid:19) ∂yj
e
hence e
∂gkq ∂yj ∂yp∂gcd∂yk ∂yq ∂xl
gjp ξ v = gab
∂yj p q (cid:20) ∂xa ∂xb ∂xl ∂xc ∂xd∂yj
e
e ee
∂yj ∂yp ∂xl ∂2yk ∂yq ∂yk ∂2yq
(3.4) +gab gcd + ξ v .
∂xa ∂xb∂yj (cid:18)∂xl∂xc∂xd ∂xc ∂xl∂xd(cid:19)(cid:21) p q
The first term of (3.4) equals to ee
∂yk ∂yj ∂gcd ∂xl ∂yk ∂gcd
gab ξ v = gab ξ v .
∂xc (cid:18) ∂xa ∂xl ∂yj(cid:19) b d (cid:18)∂xc(cid:19)(cid:18) ∂xa (cid:19) b d
Changing indexes b to p, d to q, and a to j we recognize the tangent bundle transformation of
∂gkq
gjp ξ v . The middle term vanishes since
∂xj p q
∂yq
(3.5) gcd v = gcdv = 0
∂xd q d
and the last term gives e
∂yk ∂2yq
(3.6) glbξ gcd v .
b ∂xc ∂xd∂xl q
The middle term in Γkpqξ v transforms as follows e
p q
∂gkp ∂yk ∂gcd ∂yk ∂gcp
gjq ξ v = gab ξ v = gjq ξ v .
∂yj p q (cid:18)∂xc(cid:19)(cid:18) ∂xa (cid:19) d b (cid:18)∂xc(cid:19)(cid:18) ∂xj (cid:19) p q
e
The other termes vanisheeby the same reason as in (3.5). The third term in Γkpqξ v in the new
p q
coordinates takes the form
∂gpq ∂yk ∂gpq ∂yk ∂2yq
(3.7) −gjk ξ v = − gjc ξ v −glb gcdξ v .
∂yj p q ∂xc (cid:18) ∂xj (cid:19) p q ∂xb c∂xd∂xl q
e
Weseethatthelaesttermferoem(3.7)iscanceledwith(3.6)(afterthechangeofeindexes). Taking
together the rest of terms, we get the desired transformation law
∂yk
Γkpqξ v = Γcpqξ v .
p q ∂xc p q
To show that Γ(ξ,v) ∈ S we takeea ceoveector ω ∈ S⊥ and calculate
x
hΓ(ξ,v),ωi = Γkpqξ v ω .
p q k
Using (3.2) and Lemma 3.2, we argue for each term of Γk(ξ v ) as it follows
p q
∂gkq ∂v ∂v
gjp ξ v ω = −gjpξ gkq qω = −gjpξ gqkω q = 0
∂xj p q k p ∂xj k p k∂xj
8 ANNAKOROLKO,IRINAMARKINA
and get hΓ(ξ,v),ωi = 0, that implies Γ(ξ,v) ∈ S.
The property Γ(ξ +ω,v) = Γ(ξ,v) for ω ∈ S⊥ follows from
x
1 ∂v ∂gpq
(3.8) Γk(ξ,v) = gjpξ gkq q +gjk ξ v
2 p ∂x ∂xj p q
(cid:16) j (cid:17)
and Lemma 3.2. (cid:3)
Analogously to sub-Riemannian situation [17] we have
Theorem 3.1. A vector field X ∈ S is a2-step bracket generator if and only ifΓ(ξ,·): S⊥ → S
is injective, where X = gξ. In particular, S satisfies the 2-step bracket generating hypothesis
if and only if Γ(ξ,·): S⊥ → S is injective for every nonzero form ξ ∈ T∗/S⊥.
Proof. In the proof we exploit the properties of different linear mappings which we defined up
to now. We have
(3.9) h[gξ,gη],vi = 2Γ(ξ,v)η
by (3.1) and (3.8). In order to show that the vector field X is a 2 step bracket generator we
must show that the vector fields [X,Y] mod S fill out all T/S (at each x) as Y varies over S.
In other words, the mapping
(3.10) [X,·] mod S : S → T/S is surjective at each x.
Since at any x the space T∗/S⊥ is canonically isomorphic to the dual to S, statement (3.10)
is equivalent to
(3.11) [gξ,g(·)]: T∗/S⊥ → T/S is surjective at each x, where X = gξ.
WenoticethatatanyxthespaceS⊥ iscanonically isomorphictothedualtoT/S. Thus(3.11)
is equivalent to
(3.12) Γ(ξ,·): S⊥ → S is injective at each x.
(cid:3)
We discussed earlier the relation between classical notion of covariant derivative and the
Christoffel symbols. The closest notion to the notion of covariant derivative is symmetrized
covariant derivative that was defined in [17]. It is natural to define the same concept on
ss-manifolds.
Definition 3.5. The symmetrized covariant derivative ▽ of a vector field Y is defined by
sym
∂Yq ∂Yk ∂gkq
(3.13) (▽ Y)kq = gkj +gqj −Yj .
sym ∂xj ∂xj ∂xj
Thus (▽ ) : T → T ×T .
sym x x x x
Lemma 3.6. ▽ is a well-defined differential operator from tensors of type (1,0) to sym-
sym
metric tensors of type (2,0). Furthermore, if Y is a vector field from S, that is Y = gξ, then
(▽ Y)kqv = 2Γk(ξ,v) for any v ∈ S⊥.
sym q
Proof. The symmetry follows from the symmetry of the cometric g. Let us show that ▽ Y,
sym
Y ∈ T, transforms as a tensor field of rank (2,0). We check how the first term of (3.13)
transforms with the following change of coordinates:
∂yk ∂yk ∂yj
Yk = Yj, gkj = gpq .
∂xj ∂xp∂xq
e e
SEMI-RIEMANNIAN GEOMETRY WITH CONSTRAINT 9
∂Yq ∂yk ∂yj ∂ ∂yq ∂xl
gkj = gαβ Ya
∂yj ∂xα∂xβ ∂xl (cid:18)∂xa (cid:19) ∂yj
e
e ∂yk ∂yj ∂xl ∂2yq ∂yk ∂yj ∂xl ∂Ya ∂yq
= gαβ Ya+gαβ
∂xα∂xβ ∂yj ∂xl∂xa ∂xα∂xβ ∂yj ∂xl ∂xa
∂yk ∂2yq ∂yk ∂Ya ∂yq
= gαl Ya+gαl .
∂xα∂xl∂xa ∂xα ∂xl ∂xa
Analogously, the second term
∂Yk ∂yq ∂2yk ∂yq ∂Ya ∂yk
gqj = gαl Ya+gαl .
∂yj ∂xα∂xl∂xa ∂xα ∂xl ∂xa
e
And the third term e
∂gkq ∂yj ∂ ∂yk ∂yq ∂xl
Yj = Yα gab
∂yj ∂xα ∂xl (cid:18) ∂xa∂xb(cid:19) ∂yj
e
e ∂yj ∂gab ∂yk ∂yq ∂xl
= Yα
∂xα ∂xl ∂xa∂xb∂yj
∂yj ∂2yk ∂yq ∂xl ∂yj ∂yk ∂2yq ∂xl
+ Yαgab + Yαgab .
∂xα ∂xl∂xa∂xb∂yj ∂xα ∂xa∂xl∂xb∂yj
After summation and necessary renaming of indexes we have
∂yk ∂yq
(▽ Y)kq = (▽ Y)ji.
sym ∂xi ∂xj sym
e
Let us show the second statement of the theorem. We assume Y = gξ ∈ S and v ∈ S⊥.
Then from (3.9) follows that
∂gpq ∂ξ
(▽ gξ)kqv = gkj ξ +gkjgpq p v
sym q ∂xj p ∂xj q
(cid:16) (cid:17)
∂gpk ∂ξ
+ gqj ξ +gqjgpk p v
∂xj p ∂xj q
(cid:16) (cid:17)
∂gkq
− gjpξ v
p ∂xj q
∂v ∂gpq
= gjpξ gkq q +gkj ξ v = 2Γk(ξ,v)
p ∂xj ∂xj p q
by (3.8) and gv = 0. (cid:3)
4. Hamiltonian system, exponential map and lengths of curves
ThedistributionS at each point xof ss-manifold M has the structureof Rm equippedwith
x
a nondegenerate metric Q of index ν. The presence of the nondegenerate metric yields the
x
following trichotomy.
Definition 4.1. A horizontal tangent vector w ∈ S is
x
spacelike if Q (w,w) > 0 or w = 0,
x
null if Q (w,w) = 0 and w 6= 0,
x
timelike if Q (w,w) < 0,
x
nonspacelike if it is either timelike or null.
10 ANNAKOROLKO,IRINAMARKINA
The set of all null vectors in S is called null-cone at x ∈ M. The category into which a
x
given tangent vector falls is called its causal character. The terminology is adapted from the
relativity theory, and particularly in the Lorentz case, null-vectors are called lightlike. For the
nice and complete presentation of the semi-Riemannian geometry see [15].
The covectors ξ(x) ∈ T∗/S⊥ receive the same causal structure according to the values of
x x
hg ξ(x),ξ(x)i. The covectors v ∈ S⊥(x) we shall call annihilators to distinguish them from the
x
null-covectors.
Definition 4.2. A horizontal tangent vector field X ∈ S is spacelike, null or timelike if at
each point x∈ M the vector X(x) is spacelike, null or timelike respectively.
Definition 4.3. A section ξ ∈ T∗/S⊥ is spacelike, null or timelike if at each point x∈ M the
covector ξ(x) is spacelike, null or timelike respectively.
Aswementionedfromthebeginning,weworkwiththespecialclassofadmissiblecurvesthat
tangent to the distribution S and that we called horizontal curves. We borrow this name from
thesub-Riemanniangeometry. We say thata horizontal curvec(s)is spacelike, nullor timelike
if the tangent vector c˙(s) is spacelike, null or timelike respectively at each point of c(s) where
it exists. We can give the definition of the spacelike, null or timelike curve using the causal
structure of the cotangent space T∗ according to the sign of hg ξ(s),ξ(s)i = Q (c˙(s),c˙(s)),
c(s) c(s)
where c˙(s) = g ξ(s). We call a horizontal curve the causal if the tangent vector c˙(s) (the
c(s)
covector ξ(s)) is nonspacelike
In the sub-Lorentzian case we also introduce (as in the classical Lorentz manifolds) the time
orientation.
Definition 4.4. A time orientation on (M,S,Q) is a continuous horizontal timelike section T
os S.
If M admits a time orientation T, then T divides all nonspacelike horizontal vectors into
two disjoint classes, called future directed and past directed. Namely, nonspacelike w ∈ S is
x
saidtobefuture(respectively past) directed ifQ (T(x),w) < 0(respectively Q (T(x),w) > 0).
x x
We assume that any considered in the article sub-Lorentzian manifold (M,S,Q) will be time
oriented.
Since g: T∗/S⊥ → S is injective the time orientation can be brought to T∗/S⊥.
Definition 4.5. The globally defined section τ ∈ T∗ such that T = gτ is time orientation on
T∗/S⊥.
The covectors from S⊥ we can consider as null-covectors.
The notion of arc length of a curve segment in Euclidean space generalizes in a natural way
to ss-manifolds. Since the term “arc length” can be misleading since, for example, a null-curve
has length zero. Therefore, we use the name “natural parameter” in stead of “arc length”.
Definition 4.6. Let c : [a,b] → M be a piecewise smooth curve segment in a ss-manifold
(M,S,Q). The natural parameter of c(s) is
b
L(c) = |Q(c˙(s),c˙(s))|1/2ds.
Z
a
As in the classical case it can be shown that
• the natural parameter is not changing under the monotone reparameterization and
• ifc(s)isacurvesegmentwith|c˙(s)|= |Q(c˙(s),c˙(s))|1/2 >0,thereisastrictlyincreasing
reparameterization function h such that γ = c(h) has |γ˙|= 1.
