Sub-Riemannian geometry and Lie groups II

41 Pages·2004·0.302 MB·English

by  Buliga M.

Sub-Riemannian geometry and Lie groups. Part II.
Curvature of metric spaces, coadjoint orbits and associated
representations
Marius Buliga
IMB
Bˆatiment MA
´
Ecole Polytechnique F´ed´erale de Lausanne
CH 1015 Lausanne, Switzerland
Marius.Buliga@epfl.ch
and
Institute of Mathematics, Romanian Academy
P.O. BOX 1-764, RO 70700
Bucure¸sti, Romania
[email protected]
This version: 08.07.2004
1 Introduction 2
2 Distances and metric profiles 4
2.1 Distances between metric spaces . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Metric profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Sub-Riemannian manifolds 6
3.1 Normal frames and privileged charts . . . . . . . . . . . . . . . . . . . . 7
3.2 The nilpotentization at a point . . . . . . . . . . . . . . . . . . . . . . . 10
4 Deformations of sub-Riemannian manifolds 12
5 Meaning of Mitchell theorem 1 13
6 Tangent bundles and dilatation structures 14
7 Curvature of a nice metric profile 17
8 Homogeneous spaces and motivation for curvature 18
8.1 Riemannian surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8.2 About Cartan method of equivalence for sub-Riemannian manifolds . . 22
8.3 Homogeneous contact 3 manifolds . . . . . . . . . . . . . . . . . . . . . 23
9 Homogeneous ensembles and deformations 26
10 What curvature is 29
11 Curvature and coadjoint orbits 30
12 Appendix: Uniform and conical groups 35
1 Introduction
This paper is the third in a series [4], [5], dedicated to the fundamentals of sub-
Riemannian geometry and its implications in Lie groups theory. We also bring to the
attention of the reader the paper [7] for the more analytical aspects of the theory.
Thepointofviewofthesepapersisthatwhatwearedealingwitharemanifestations
of the emerging non-Euclidean analysis. This become visible especially in the metric
study of Lie groups endowed with left invariant generating distributions.
Metric profiles and their curvature first appeared in [6]. They give new insights
into the behaviour of metric spaces. The subject is in close link with sub-Riemannian
Lie groups because curvatures could be classified by comparison with (metric profiles
of) homogeneous spaces. The homogeneous spaces we are interested in can be seen as
factor spaces of Lie groups with left invariant distributions.
The curvature of a metric space in a point is, by definition, the rectifiability class
of the metric profile associated to the point. There are two problems here that we are
trying to solve.
The first problem comes from the fact that to a metric space with some geometric
structure we can associate several metric profiles. The most interesting are the profile
as a metric space and the dilatation profile, associated with the dilatation structure of
the space.
The dilatation structure is the basic object in the study of differential properties
of metric spaces of a certain type. (For example any Riemannian or sub-Riemannian
manifold,withorwithoutconicalsingularities,hasadilatationstructure. Butthereare
metricspaceswhicharenotadmittingmetrictangentspaces, buttheyadmitdilatation
structures). This structure tells us what is the good notion of analysis on that space.
There is an infinite class of different such analysis and the classical one, which we call
Euclidean, is only one of them.
Toeachmetricprofileofthespaceweassociateacurvature. Weshallhavetherefore
a metric curvature and a dilatation curvature. The metric curvature is more difficult
to describe and in fact the dilatation curvature contains more complex informations.
The first problem related to curvature is: what curvature we measure?
Thesecondproblemofcurvatureisrelatedtothefactthatthedefinitionofcurvature
as a rectifiability class is too abstract to work with, unless we have a ”good” set of
representatives for the rectifiability classes. Suppose that we have made such a choice
of the ”good” class of representatives. Then we shall say that the curvature of a space
inapointistherepresentantoftherectifiabilityclass. Thesecondproblemofcurvature
is: what is a ”good” class of representatives?
This is obviously a subjective matter; anyway in this paper this subjectivity is
revealed and, once revealed, the proposed choices become less subjective.
The goal of this paper is to show that curvature in the sense of rectifiability class
of the dilatation profile can be classified using coadjoint orbits representations. We
establish here a bridge between:
- curvature notion in spaces which admit metric tangent spaces at any point, and
- self-adjoint representations of algebras naturally associated with the structure of
the tangent space.
In our opinion this is the second link between sub-Riemannian geometry and quan-
tum mechanics. The first link can be uncovered from Buliga [5] section 5 ”Case of the
Heisenberg group”, as explained in the section 11 of the present paper.
An appendix concerning uniform and conical groups is added to the paper. More
information about the subject can be found in Buliga [5], sections 3 and 4.
2 DISTANCES AND METRIC PROFILES 4
2 Distances and metric profiles
The references for the first subsection are Gromov [12], chapter 3, Gromov [10], and
Burago & al. [8] section 7.4. There are several definitions of distances between metric
spaces. The very fertile idea of introducing such distances belongs to Gromov.
2.1 Distances between metric spaces
InordertointroducetheHausdorffdistancebetweenmetricspaces,recalltheHausdorff
distance between subsets of a metric space.
Definition 2.1 For any set A ⊂ X of a metric space and any ε > 0 set the ε neigh-
bourhood of A to be
[
A = B(x,ε)
ε
x∈A
The Hausdorff distance between A,B ⊂ X is defined as
dX(A,B) = inf{ε > 0 : A ⊂ B , B ⊂ A }
H ε ε
ByconsideringallisometricembeddingsoftwometricspacesX,Y intoanarbitrary
metric space Z we obtain the Hausdorff distance between X, Y (Gromov [12] definition
3.4).
Definition 2.2 The Hausdorff distance d (X,Y) between metric spaces X Y is the
H
infimum of the numbers
dZ(f(X),g(Y))
H
for all isometric embeddings f : X → Z, g : Y → Z in a metric space Z.
If X, Y are compact then d (X,Y) < +∞.
H
TheHausdorffdistancebetweenisometricspacesequals0. Theconverseisalsotrue
in the class of compact metric spaces (Gromov op. cit. proposition 3.6).
Likewise one can think about a notion of distance between pointed metric spaces.
A pointed metric space is a triple (X,x,d), with x ∈ X. Gromov [10] introduced the
distance between pointed metric spaces (X,x,d ) and (Y,y,d ) to be the infimum of
X Y
all ε > 0 such that there is a distance d on the disjoint sum X ∪Y, which extends the
distances on X and Y, and moreover
- d(x,y) < ε,
- the ball B(x, 1) in X is contained in the ε neighbourhood of Y,
ε
- the ball B(y, 1) in Y is contained in the ε neighbourhood of X.
ε
Denote by [X,x,d ] the isometry class of (X,x,d ), that is the class of spaces
X x
(Y,y,d ) such that it exists an isometry f : X → Y with the property f(x) = y.
Y
The Gromov distance between isometry classes of pointed metric spaces is almost
a distance, in the sense that whenever two of the spaces [X,x,d ], [Y,y,d ], [Z,z,d ]
X Y Z
have diameter at most equal to 2, then the triangle inequality for this distance is true.
We shall use this distance and the induced convergence for isometry classes of the form
[X,x,d ], with diam X ≤ 2.
X
2 DISTANCES AND METRIC PROFILES 5
2.2 Metric profiles
The notion of metric profile was introduced in Buliga [6].
We shall denote by CMS the set of isometry classes of pointed compact metric
spaces. The distance on this set is the Gromov distance between (isometry classes of)
pointed metric spaces and the topology is induced by this distance.
To any locally compact metric space we can associate a metric profile.
Definition 2.3 The metric profile associated to the locally metric space (M,d) is the
assignment (for small enough ε > 0)
1
(ε > 0, x ∈ M) 7→ Pm(ε,x) = [B¯(x,1),x, d] ∈ CMS
ε
We can define a notion of metric profile regardless to any distance.
Definition 2.4 A metric profile is a curve P : [0,a] → CMS such that:
(a) it is continuous at 0,
(b) for any b ∈ [0,a] and fixed ε ∈ (0,1] we have
d (P(εb),Pm(ε,x)) = O(b)
GH d
b
We used here the notation P(b) = [B¯(x,1),d ] and Pm(ε,x) = [B¯(x,1), 1d ].
b db ε b
The metric profile is nice if
d (P(εb),Pm(ε,x)) = O(bε)
GH d
b
The metric profile ε 7→ Pm(ε,x) of a metric space (M,d) for a fixed x ∈ M is a
metric profile in the sense of the definition 2.4 if and only if the space (M,d) admits a
tangent cone. Indeed, a tangent cone [V,v,d ] exists if and only if the following limit
v
[V,v,d ] = limPm(ε,x)
v
ε→0
exists. In this case the metric profile Pm(·,x) can be prolonged to ε = 0. The prolon-
gation is a metric profile in the sense of definition 2.4. Indeed, we have still to check
the property (b). But this is trivial, because for any ε,b > 0, sufficiently small, we have
Pm(εb,x) = Pm(ε,x)
d
b
where d = (1/b)d and Pm(ε,x) = [B¯(x,1), 1d ].
Notebthat in the definitdibon of a nice metriεc pbrofile is not stated that P(0) = Pm(0).
d
b
The metric profile of a Riemannian homogeneous space is just a curve in the space
CMS, continuous at 0. Likewise, if we look at a homogeneous regular sub-Riemannian
manifold, the metric profile is not depending on points in the manifold.
3 SUB-RIEMANNIAN MANIFOLDS 6
3 Sub-Riemannian manifolds
Classical references to this subject are Bella¨ıche [2] and Gromov [11]. The interested
reader is advised to look also to the references of these papers.
Intheliteratureonsub-Riemannianmanifoldsnoteverythingwrittencanbetrusted.
The source of errors lies in the first notions and constructions, mostly in the fact that
obvious properties connected to a Riemannian manifold are not true (or yet unproven)
for a sub-Riemannian manifold. Of special importance is the difference between a nor-
mal frame and the frame induced by a privileged chart. Also, there are many things in
the sub-Riemannian realm which have no correspondent in the Riemannian case.
To close this introductory comments, let us remark that for the informed reader
should be clear that in the sub-Riemannian realm there are problems even with the
notion of manifold. Everybody agrees to define a sub-Riemannian manifold as in the
definition3.2. However, abetternamefortheobjectinthisdefinitionwouldbe”model
of sub-Riemannian manifold”. We can illustrate this situation by the following com-
parison: the notion of sub-Riemannian manifold given in the definition 3.2 is to the
real notion what the Poincar´e disk is to the hyperbolic plane. Only that apparently
nobody found the real notion yet.
LetM beaconnectedmanifold. Adistribution(orhorizontalbundle)isasubbundle
D of M. To any point x ∈ M there is associated the vectorspace D ⊂ T M.
x x
Given the distribution D, a point x ∈ M and a sufficiently small open neighbour-
hood x ∈ U ⊂ M, one can define on U a filtration of bundles as follows. Define first
the class of horizontal vectorfields on U:
X1(U,D) = {X ∈ Γ∞(TU) : ∀y ∈ U , X(y) ∈ D }
y
Next, define inductively for all positive integers k:
Xk+1(U,D) = Xk(U,D)∪[X1(U,D),Xk(U,D)]
Here [·,·] denotes vectorfields bracket. We obtain therefore a filtration Xk(U,D) ⊂
Xk+1(U,D). Evaluate now this filtration at x:
n o
Vk(x,U,D) = X(x) : X ∈ Xk(U,D)
Therearem(x),positiveinteger,andsmallenoughU suchthatVk(x,U,D) = Vk(x,D)
for all k ≥ m and
D = V1(x,D) ⊂ V2(x,D) ⊂ ... ⊂ Vm(x)(x,D)
x
We equally have
ν (x) = dimV1(x,D) < ν (x) = dimV2(x,D) < ... < n = dimM
1 2
Generally m(x), ν (x) may vary from a point to another.
k
The number m(x) is called the step of the distribution at x.
3 SUB-RIEMANNIAN MANIFOLDS 7
Definition 3.1 The distribution D is regular if m(x), ν (x) are constant on the man-
k
ifold M.
The distribution is completely non-integrable if for any x ∈ M we have Vm(x) =
T M.
x
Definition 3.2 A sub-Riemannian (SR) manifold is a triple (M,D,g), where M is
a connected manifold, D is a completely non-integrable distribution on M, and g is a
metric (Euclidean inner-product) on the distribution (or horizontal bundle) D.
A horizontal curve c : [a,b] → M is a curve which is almost everywhere derivable
and for almost any t ∈ [a,b] we have
c˙(t) ∈ D
c(t)
The class of horizontal curves will be denoted by Hor(M,D).
The lenght of a horizontal curve is
Z b
p
l(c) = g(c(t))(c˙(t),c˙(t)) dt
a
The length depends on the metric g.
The Carnot-Carath´eodory (CC) distance associated to the sub-Riemannian mani-
fold is the distance induced by the length l of horizontal curves:
d(x,y) = inf{l(c) : c ∈ Hor(M,D) , c(a) = x , c(b) = y}
The Chow theorem ensures the existence of a horizontal path linking any two suf-
ficiently closed points, therefore the CC distance it at least locally finite.
Weshallworkfurtheronlywithregularsub-Riemannianmanifolds, ifnototherwise
stated.
3.1 Normal frames and privileged charts
Bella¨ıche introduced the concept of privileged chart around a point x ∈ M.
Let(x ,...,x ) 7→ φ(x ,...,x ) ∈ M beachartofM aroundx(i.e. xhascoordinates
1 n 1 n
(0,....,0)). Denote by X ,...,X the frame of vectorfields associated to the coordinate
1 n
chart. The chart is called adapted at x if the following happens: X (x),...,X (x) form
1 ν1
a basis of V1(x,D), X (x),...,X (x) span a space in V2(x,D) which is transversal
ν1+1 ν2
to V1(x,D) and X (x),...,X (x) form a basis of V2(x,D), and so on.
1 ν2
Suppose that the frame X ,...,X corresponds to a chart adapted at x. The degree
1 n
of X at x is then k if X ∈ Vk \Vk−1.
i i
Definition 3.3 A chart is privileged around the point x ∈ M if it is adapted at x and
for any i = 1,...,n the function
t 7→ d(x,φ(...,t,...))
(with t on the position i) is exactly of order deg X at t = 0.
i
3 SUB-RIEMANNIAN MANIFOLDS 8
Privileged charts always exist, as proved by Bella¨ıche [2] Theorem 4.15. As an
example consider a Lie group G endowed with a left invariant distribution. The distri-
bution is completely non-integrable if it is generated by the left translation of a vector
subspace D of the algebra g = T G which bracket generates the whole algebra g. Then
e
the exponential map is a privileged chart at the identity e ∈ G, but generically not
privileged at x 6= e.
Let X be a vectorfield on M and x ∈ M. The degree of X at x is the order of the
function
t ∈ R → R ,t 7→ d(x,exp(tX)(x))
and it is denoted by deg X. The vectorfield is called regular in an open set U ⊂ M if
x
deg X is constant for all x ∈ U.
x
Consider now a frame of vectorfields X ,...,X defined in U ⊂ M, open set. Let
1 n
x ∈ U and define:
V (x) = span {X (x) : deg X = i}
i k x k
Definition 3.4 A frame X ,...,X , defined in an open set U ⊂ M, is normal if all
1 n
vectorfields X are regular in U and moreover at any x ∈ U and any i = 1,...,n one
k
has:
Vi(x) = V (x)+...+V (x)
1 i
(direct sum).
A normal frame transforms the filtration into a direct sum. Each tangent space
decomposes as a direct sum of vectorspaces V . Moreover, each space Vi decomposes
i
in a direct sum of spaces V with k ≤ i.
k
Normal frames exist. Indeed, start with a frame X ,...,X such that for any x
1 r
X (x),...,X (x) form a basis for D(x). Associate now to any word a ....a with letters
1 r 1 q
in the alphabet 1,...,r the multi-bracket
[X ,[...,X ]...]
a1 aq
Onecanadd,inthelexicographicorder,elementstotheset{X ,...,X },untilanormal
1 r
frame is obtained.
To a normal frame X ,...,X and the point x ∈ U one can associate a privileged
1 n
chart. Inded, such a chart is defined by:
n !
X X
(a ,...,a ) ∈ Rn ≡ T M 7→ φ ( a X (x)) = exp a X (x) (3.1.1)
1 n x x i i i i
i=1
Remark how the privileged chart changes with the base-point x.
The intrinsic dilatations associated to a normal frame, in a point x, are defined via
a choice of a privileged chart based at x. In such a chart φ, for any ε > 0 (sufficiently
small if necessary) the dilatation is defined by
δ (x ) = (εdeg ix )
ε i i
3 SUB-RIEMANNIAN MANIFOLDS 9
With the use of the privileged charts (3.1.1), for any ε > 0 (sufficiently small if neces-
sary) the dilatations are
n ! ! n !
X X
δx exp a X (x) = exp a εdegXiX (x)
ε i i i i
i=1 i=1
In terms of vectorfields, one can use an intrinsic dilatation associated to the normal
frame. This dilatation transforms X into
i
∆ X = εdeg XiX
ε i i
One can define then, as in Gromov [11], section 1.4, or Vodop’yanov [24], deformed
vectorfields with respect to fixed x by
(cid:16) (cid:17)
Xˆx(ε)(y) = ∆ (δx)−1∗X (y)
i ε ε i
When ε → 0 the vectorfields Xˆx(ε) converge uniformly to a vectorfield XN,x, on
i i
small enough compact neighbourhoods of x.
The nilpotentization of the distribution with respect to the chosen normal frame,
in the point x, is then the bracket
[X,Y]x = lim[Xˆx(ε),Yˆx(ε)] (3.1.2)
N
ε→0
We have the equality
[X ,X ]x (x) = [XN,x,XN,x](x)
i j N i j
It is generically false that there are privileged coordinates around an open set in
M. We can state this as a theorem.
Theorem 3.5 Let (M,D,g) be a regular sub-riemannian manifold of topological di-
mension n. There are ∅ =6 U ⊂ M an open subset and φ : B ⊂ Rn → U such that φ is
a privileged chart for any x ∈ U if and only if (M,D,g) is a Riemannian manifold.
Proof. If (M,D,g) is Riemannian then it is known that such privileged charts exist.
We have to prove the converse. Suppose there is a map φ : B → U, B open set in Rn,
φ surjective, such that for any x ∈ U φ is privileged. Consider the frame X ,...,X of
1 n
vectorfields tangent to coordinate lines induced by φ. Then this is a normal frame in
U. Moreover
[X ,X ] = 0
i j
for any i,j = 1,...,n therefore the nilpotentization bracket (3.1.2) in any point x ∈ U
is equal to 0. According to Mitchell [18] theorem 1 (in this paper theorem 5.1 section
5) the tangent cone in the metric sense to x is the Euclidean Rn. But this implies that
the manifold (which is supposed regular) is Riemannian. (cid:4)
3 SUB-RIEMANNIAN MANIFOLDS 10
3.2 The nilpotentization at a point
The nilpotentization defines at point x a nilpotent Lie algebra structure. This in turn
gives a nilpotent group operation on the tangent space at x (in the classical sense),
by the use of the Baker-Campbell-Hausdorff formula. Denote this group N(x). By
constructionthegroupN(x)anditsLiealgebraareidenticalassets(otherwisesaidthe
exponential is the identity). What we have is a vectorspace endowed with a Lie bracket
and a multiplication operation, linked by the Baker-Campbell-Hausdorff formula.
ConsiderthedistributionND(x)onthegroupN(x)whichisobtainedbylefttrans-
lation in N(x) of the vectorspace D ⊂ N(x) (inclusion of a space in the Lie algebra
x
of the group N(x), seen as the tangent space to the neutral element of the group). We
equally transport the metric g(x) by nilpotent left translations in N(x) and we denote
the metric on the bundle ND(x) by g (x). We obtain a regular sub-Riemannian man-
N
ifold (N(x),ND(x),g (x)). The associated Carnot-Carath´eodory distance is denoted
N
by dx .
N
As a metric space (N(x),dx ) is a cone based in the neutral element of N(x). It is
N
therefore a good candidate for being the metric tangent cone to the space (M,d) at x.
Endowed with the group operation and dilatations, N(x) is a Carnot group.
Definition 3.6 A Carnot (or stratified nilpotent) group is a connected simply con-
nected group N with a distinguished vectorspace V such that the Lie algebra of the
1
group has the direct sum decomposition:
m
X
n = V , V = [V ,V ]
i i+1 1 i
i=1
The number m is the step of the group. The number
m
X
Q = i dimV
i
i=1
is called the homogeneous dimension of the group.
Any Carnot group admits a one-parameter family of dilatations. For any ε > 0, the
associated dilatation is:
m m
X X
x = x 7→ δ x = εix
i ε i
i=1 i=1
Any such dilatation is a group morphism and a Lie algebra morphism.
We collect some important facts to be known about Carnot groups:
(a) The metric topology and uniformity of N are the same as Euclidean topology
and uniformity respective.
(b) The ball B(0,r) looks roughly like the box (cid:8)x = Pm x : kx k ≤ ri(cid:9).
i=1 i i