Lecture notes on sub-Riemannian geometry

186 Pages·2017·1.373 MB·English

by  Le Donne E

Lecture notes on
sub-Riemannian geometry
by Enrico Le Donne.
http://enrico.ledonne.googlepages.com/
Version of Summer 2017

Contents
0 A Brief Introduction* 1
0.1 Sub-Riemannian geometries as models . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.1.1 Many examples from Mathematics . . . . . . . . . . . . . . . . . . . . . . . . 5
Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Classical mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Symplectic and contact geometry . . . . . . . . . . . . . . . . . . . . . . . . . 5
Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Diffusion on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Analysis of hypoelliptic operators . . . . . . . . . . . . . . . . . . . . . . . . . 5
Geometric Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Cauchy-Riemann (or CR) geometry . . . . . . . . . . . . . . . . . . . . . . . 6
Univalent Function Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.1.2 Many examples from Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
The geometry of principal bundles with connections . . . . . . . . . . . . . . 7
Falling cats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
From mechanics: parking cars, rolling balls, moving robots, and satellites . . 8
Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Quantum mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Even more examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1 The Main Example: the Heisenberg Group 13
1.1 An isoperimetric problem on the plane . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 The contact-geometry formulation of the problem . . . . . . . . . . . . . . . . . . . . 14
1.3 The Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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1.3.1 The 3D nilpotent non-Abelian matrix group . . . . . . . . . . . . . . . . . . . 18
1.3.2 The uniqueness of the Heisenberg algebra . . . . . . . . . . . . . . . . . . . . 19
1.4 The subRiemannian Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.1 The geodesics and the spheres in the Heisenberg group . . . . . . . . . . . . . 20
1.4.2 Dilations on the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . 23
The dimension of the Heisenberg group . . . . . . . . . . . . . . . . . . . . . 25
More pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 The General Theory 29
2.1 A review of metric geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.2 Length of curves in metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.3 Length Space, Intrinsic Metrics, and Geodesic Spaces . . . . . . . . . . . . . 32
2.1.4 Isometries and Lipschitz maps . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.5 Hausdorff Measures and Dimension . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 A review of differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.1 Vector fields and Lie brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.2 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.3 Riemannian and Finsler geometry . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 The Definition of Carnot-Carath´eodoryDistances . . . . . . . . . . . . . . . . . . . . 42
2.3.1 Bracket-generatingdistributions . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.2 SubFinsler structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.3 The generalization of Control Theory . . . . . . . . . . . . . . . . . . . . . . 44
2.3.4 The general definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.5 Energy VS Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Chow’s Theorem and existence of geodesics . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.1 Reachable sets of bracket-generatingdistributions . . . . . . . . . . . . . . . 47
2.4.2 The metric version of Chow’s theorem . . . . . . . . . . . . . . . . . . . . . . 48
2.4.3 Comparison of length structures . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.4 Existence of geodesics in CC spaces . . . . . . . . . . . . . . . . . . . . . . . 50
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2.5 Equiregular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Ball-Box Theorem and Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.1 Ball-Box Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.2 Dimensions of CC spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.6.3 The problem of dimensions of submanifolds in CC spaces . . . . . . . . . . . 55
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Nilpotent Groups and Carnot Groups 59
3.1 A review of Lie groups and their Lie algebras . . . . . . . . . . . . . . . . . . . . . . 59
3.1.1 Exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.2 Nilpotent Lie groups and nilpotent Lie algebras . . . . . . . . . . . . . . . . . 63
3.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.1.4 The BCH formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.5 Matrix groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1.6 Adjoint operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1.7 Simply connected nilpotent Lie groups . . . . . . . . . . . . . . . . . . . . . . 68
3.1.8 Homogeneous manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2 Stratified Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.1 Some elementary facts about stratifications . . . . . . . . . . . . . . . . . . . 76
3.2.2 Dilation Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 Carnot groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.1 Dilations and CC distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 A deeper study of Carnot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.1 A good basis for a Carnot group . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.2 Local-to-globalusing dilations, and canonical coordinates . . . . . . . . . . . 83
3.4.3 A direct, effective proof of Chow’s Theorem . . . . . . . . . . . . . . . . . . . 85
The brackets as products of exponentials . . . . . . . . . . . . . . . . . . . . 85
3.4.4 A proof of the Ball-Box Theorem for Carnot groups . . . . . . . . . . . . . . 89
3.4.5 Haar, Hausdorff and Lebesgue measures . . . . . . . . . . . . . . . . . . . . . 91
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
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4 Limits of Riemannian and subRiemannian manifolds 97
4.1 Limits of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1.1 A topology on the space of metric spaces . . . . . . . . . . . . . . . . . . . . 97
4.1.2 Asymptotic cones and tangent spaces . . . . . . . . . . . . . . . . . . . . . . 98
4.2 Asymptotic cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.1 SubRiemannian Carnot group as Riemannian limits . . . . . . . . . . . . . . 99
4.2.2 More general limits of Riemannian manifolds . . . . . . . . . . . . . . . . . . 99
4.2.3 Preparatoryexample: The Riemannian Heisenberg group . . . . . . . . . . . 100
4.2.4 Towardthe general setting: Gronwall Lemma . . . . . . . . . . . . . . . . . . 101
4.2.5 Asymptotic cones of Riemannian stratified groups . . . . . . . . . . . . . . . 103
4.3 Tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3.1 Preparatoryexample: The subRiemannian rototranslationgroup . . . . . . . 104
Quantitative Chow’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3.2 Nilpotization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.3 Mitchell’s Theorem on tangent cones . . . . . . . . . . . . . . . . . . . . . . . 109
4.4 A metric characterizationof Carnot groups . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.1 Proof of the characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5 Visual boundary of complex hyperbolic space* 113
5.1 Complex hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 CAT(-1) spaces and visual boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3 Heisenberg group as boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Analysis on Nilpotent Groups 115
6.1 Rademacher Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.1 Margulis-Mostow’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.2 Pansu’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Preliminaries to the proof of Pansu’s theorem . . . . . . . . . . . . . . . . . . 116
Proof of Pansu’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.1.3 Applications to non-embeddability . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Regularity problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
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Comments regarding geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Comments regarding metric spheres . . . . . . . . . . . . . . . . . . . . . . . 123
Comments regarding minimal surfaces and isoperimetric solutions . . . . . . 123
6.2.1 Common general philosophical strategy for regularity . . . . . . . . . . . . . 124
6.3 Generalized hyper-surfaces: sets with finite perimeter. . . . . . . . . . . . . . . . . . 124
6.3.1 A review of divergence and distributions . . . . . . . . . . . . . . . . . . . . . 124
6.3.2 Caccioppoli sets: sets of locally finite perimeter . . . . . . . . . . . . . . . . . 126
6.3.3 Notions of rectificability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3.4 Notions of surface measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 Partial regularity results and open questions . . . . . . . . . . . . . . . . . . . . . . . 129
6.4.1 Results on geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4.2 Results on metric spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.4.3 Results on the isoperimetric problem . . . . . . . . . . . . . . . . . . . . . . . 131
6.4.4 Results on minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.4.5 More results on regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.5 Translations and flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5.1 X-derivative of nice functions and domains . . . . . . . . . . . . . . . . . . . 136
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Nilpotent Discrete Groups 139
7.1 Elements of Geometric Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2 The growth rate of balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2.1 Invariance of the growth rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2.2 Polynomial growth and virtual nilpotency . . . . . . . . . . . . . . . . . . . . 142
7.3 Asymptotic cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.4 The Malcev closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.5 The limit CC metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A Equivalent Definitions of Length 147
B Dido’s Problem 151
B.1 A proof of the isoperimetric problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
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B.1.1 The variation of length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.1.2 The area functional and its variation . . . . . . . . . . . . . . . . . . . . . . . 152
B.1.3 The conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
C Curves in sub-Finsler Nilpotent Groups 157
C.0.1 A special sub-Finsler geometry on nilpotent groups . . . . . . . . . . . . . . . 157
The “development” of a curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 158
C.0.2 Horizontal lines as geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
C.0.3 Lifts of curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
C.0.4 Curves in free nilpotent Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 164
C.0.5 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
D Parking motorbikes and cars* 167
Bibliography 168
Index 176
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Chapter 0
A Brief Introduction*
The asterisk * will denote incompleteness of the chapter or section.
Thismanuscriptismostlybasedonthefollowingbooksandpapers: [Pon66,War83,CG90,Bel96,
Gro99, AFP00, BBI01, Hel01, Mon02, AKL09, Kna02, ?, HN12, BrL13, LD15]. It has been written
initially for a course entitled ‘Sub-Riemannian Geometry’ which was taught once at ETH in Zu¨rich
(Switzerland) during Fall 2009 and then at the University of Jyva¨skyla¨ (FInland) in Spring 2014.
Consequently, some parts were added after the author thought a course entitled ‘Carnot groups’ at
a summer school in Levico Terme (Trento, Italy) in 2015 and a course entitled ‘Riemannian and
subRiemanniangeometryonLiegroups’atthesummerschoolinNeurogeometryin2017inCortona
(Italy).
Sub-RiemanniangeometryisageneralizationofRiemanniangeometry. Roughlyspeaking,asub-
RiemannianmanifoldisaRiemannianmanifoldtogetherwithaconstrainonadmissibledirectionof
movements. In Riemannian geometry any (smoothly embedded) curve has locally finite length. In
sub-Riemannian geometry, if the curve fails to satisfy the obligation of the constrain, then it would
have infinite length.
Oneclassicalexample oneshouldcarryinmind is comingfrommechanics. Indeed, the statiofa
moving object are enclosedby its position in space and the speeds of its parts: the momenta. Thus
in the manifold ‘positions times speeds’ the possible evolutions of the object should satisfy the fact
that the derivatives of the first coordinates are equal the second coordinates. In particular, some
trajectoriesarenotallowed. Astrivialexamples,youcannotvaryyourspeedwithoutchangingyour
position or, similarly, you cannot move into another place at speed zero!
The 3D Heisenberg group is the most important sub-Riemannian geometry that is not in fact
1
0-ABrief Introduction* February 17, 2018
z
y
x
Figure 1: A standard distribution on R3
a Riemannian one. It is also not difficult to visualize some of its features. Topologically it is R3.
The constrain on curves is given by what is called a ‘distribution of planes’. Similarly as a smooth
vectorfield smoothlyassignsa tangentvectorateachpointofthe manifold,a distributionofplanes
smoothlyassignstoeachpointaplane (inside the 3Dtangentspace). The curvesthatwillbe called
‘admissible’ will be those curves that are tangent to such a distribution.
The great feature of the Heisenberg group is that its distribution is curly enough in a way that
eachpairofpointcanbeconnectedbyatleastoneadmissiblecurve. Fromthisfactonecandefinea
(finite) distance similarly to the Riemannian case: the distance between two points p and q is given
by the infimum of the length of all those admissible curves from p to q,
d(p,q)=inf Length(γ) : γ admissible, from p to q . (0.0.1)
{ }
In the first part of this lecture notes we will discuss the following facts:
1) Such a distance d turns the space R3 into a metric space with the same standard topology.
2) For any two points there is in fact a geodesic curve, i.e., the distance of the points equals the
lengthofa curve;the readercouldreallythink that the length ofsuchcurveis its Euclideanlength.
3) This metric space is really new: it is not Riemannian. It is not evenbiLipschitz equivalent to
a Riemannian distance. In fact, the Heisenberg geometryresemble fractalgeometry. Indeed, such a
metric on this topologically 3-dimensional object will have metric (i.e., Hausdorff) dimension equal
to 4.
2