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El. knyga: Comprehensive Introduction to Sub-Riemannian Geometry

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(Université de Paris VII (Denis Diderot)), (Centre National de la Recherche Scientifique (CNRS), Paris), (Scuola Internazionale Superiore di Studi Avanzati, Trieste)
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Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. In the last two decades sub-Riemannian geometry has emerged as an independent research domain impacting on several areas of pure and applied mathematics, with applications to many areas such as quantum control, Hamiltonian dynamics, robotics and Lie theory. This comprehensive introduction proceeds from classical topics to cutting-edge theory and applications, assuming only standard knowledge of calculus, linear algebra and differential equations. The book may serve as a basis for an introductory course in Riemannian geometry or an advanced course in sub-Riemannian geometry, covering elements of Hamiltonian dynamics, integrable systems and Lie theory. It will also be a valuable reference source for researchers in various disciplines.

This comprehensive introduction to sub-Riemannian geometry proceeds from classical topics to cutting-edge theory and applications. The only prerequisites are calculus, linear algebra and differential equations. It can be used for graduate courses in Riemannian or sub-Riemannian geometry, or as a reference for researchers in several disciplines.

Recenzijos

'It is wonderful to have a wide swath of the work of this school explained clearly and set down in one place. I am understanding some of the concepts described for the first time. I am grateful to the three authors for their efforts in putting this book together.' Richard Montgomery, Bulletin of the American Mathematical Society 'This textbook is a valuable reference in sub-Riemannian geometry, providing a systematic and firm foundation to the theory It is my opinion that this textbook will serve as a solid reference for many researchers in the field, and will contribute to the development of the subject in the forthcoming years.' Luca Rizzi, Mathematical Reviews 'The book can be used for either an introductory or advanced course on sub-Riemannian geometry (the authors suggest which chapters to use for each case), but it also constitutes a state-of-the-art reference for most of the topics that it treats and will be an essential work for researchers active in sub-Riemannian geometry.' Robert Neel, MAA Reviews ' an excellent resource for a broad range of topics in Riemannian and sub-Riemannian geometry. I am strongly convinced that this will become one of the main references for people interested in these topics, ranging from students to specialists.' Alpįr R. Mészįros, zbMATH

Daugiau informacijos

Provides a comprehensive and self-contained introduction to sub-Riemannian geometry and its applications. For graduate students and researchers.
Preface xvii
Introduction 1(10)
1 Geometry of Surfaces in R3
11(34)
1.1 Geodesies and Optimality
11(8)
1.1.1 Existence and Minimizing Properties of Geodesies
16(2)
1.1.2 Absolutely Continuous Curves
18(1)
1.2 Parallel Transport
19(4)
1.2.1 Parallel Transport and the Levi-Civita Connection
20(3)
1.3 Gauss-Bonnet Theorems
23(13)
1.3.1 Gauss-Bonnet Theorem: Local Version
23(4)
1.3.2 Gauss-Bonnet Theorem: Global Version
27(4)
1.3.3 Consequences of the Gauss-Bonnet Theorems
31(2)
1.3.4 The Gauss Map
33(3)
1.4 Surfaces in R3 with the Minkowski Inner Product
36(4)
1.5 Model Spaces of Constant Curvature
40(4)
1.5.1 Zero Curvature: The Euclidean Plane
40(1)
1.5.2 Positive Curvature: The Sphere
41(2)
1.5.3 Negative Curvature: The Hyperbolic Plane
43(1)
1.6 Bibliographical Note
44(1)
2 Vector Fields
45(22)
2.1 Differential Equations on Smooth Manifolds
45(6)
2.1.1 Tangent Vectors and Vector Fields
45(2)
2.1.2 How of a Vector Field
47(1)
2.1.3 Vector Fields as Operators on Functions
48(1)
2.1.4 Nonautonomous Vector Fields
49(2)
2.2 Differential of a Smooth Map
51(2)
2.3 Lie Brackets
53(4)
2.4 Frobenius' Theorem
57(3)
2.4.1 An Application of Frobenius' Theorem
59(1)
2.5 Cotangent Space
60(2)
2.6 Vector Bundles
62(2)
2.7 Submersions and Level Sets of Smooth Maps
64(2)
2.8 Bibliographical Note
66(1)
3 Sub-Riemannian Structures
67(42)
3.1 Basic Definitions
67(13)
3.1.1 The Minimal Control and the Length of an Admissible Curve
71(3)
3.1.2 Equivalence of Sub-Riemannian Structures
74(2)
3.1.3 Examples
76(1)
3.1.4 Every Sub-Riemannian Structure is Equivalent to a Free Sub-Riemannian Structure
77(3)
3.2 Sub-Riemannian Distance and Rashevskii-Chow Theorem
80(7)
3.2.1 Proof of the Rashevskii-Chow Theorem
81(5)
3.2.2 Non-Bracket-Generating Structures
86(1)
3.3 Existence of Length-Minimizers
87(10)
3.3.1 On the Completeness of the Sub-Riemannian Distance
89(2)
3.3.2 Lipschitz Curves with respect to d vs. Admissible Curves
91(2)
3.3.3 Lipschitz Equivalence of Sub-Riemannian Distances
93(1)
3.3.4 Continuity of d with respect to the Sub-Riemannian Structure
94(3)
3.4 Pontryagin Extremals
97(7)
3.4.1 The Energy Functional
99(1)
3.4.2 Proof of Theorem 3.59
100(4)
3.5 Appendix: Measurability of the Minimal Control
104(2)
3.5.1 A Measurability Lemma
104(2)
3.5.2 Proof of Lemma 3.12
106(1)
3.6 Appendix: Lipschitz vs. Absolutely Continuous Admissible Curves
106(2)
3.7 Bibliographical Note
108(1)
4 Pontryagin Extremals: Characterization and Local Minimality
109(40)
4.1 Geometric Characterization of Pontryagin Extremals
109(7)
4.1.1 Lifting a Vector Field from M to T*M
110(1)
4.1.2 The Poisson Bracket
111(3)
4.1.3 Hamiltonian Vector Fields
114(2)
4.2 The Symplectic Structure
116(3)
4.2.1 Symplectic Form vs. Poisson Bracket
117(2)
4.3 Characterization of Normal and Abnormal Pontryagin Extremals
119(8)
4.3.1 Normal Extremals
120(4)
4.3.2 Abnormal Extremals
124(2)
4.3.3 Codimension-1 and Contact Distributions
126(1)
4.4 Examples
127(9)
4.4.1 2D Riemannian Geometry
128(2)
4.4.2 Isoperimetric Problem
130(4)
4.4.3 Heisenberg Group
134(2)
4.5 Lie Derivative
136(2)
4.6 Symplectic Manifolds
138(2)
4.7 Local Minimality of Normal Extremal Trajectories
140(8)
4.7.1 The Poincare-Cartan 1-Form
140(2)
4.7.2 Normal Pontryagin Extremal Trajectories are Geodesies
142(6)
4.8 Bibliographical Note
148(1)
5 First Integrals and Integrable Systems
149(22)
5.1 Reduction of Hamiltonian Systems with Symmetries
149(4)
5.1.1 An Example of Symplectic Reduction: the Space of Affine Lines in R"
152(1)
5.2 Riemannian Geodesic Flow on Hypersurfaces
153(4)
5.2.1 Geodesies on Hypersurfaces
153(1)
5.2.2 Riemannian Geodesic Flow and Symplectic Reduction
154(3)
5.3 Sub-Riemannian Structures with Symmetries
157(2)
5.4 Completely Integrable Systems
159(4)
5.5 Arnold-Liouville Theorem
163(3)
5.6 Geodesic Flows on Quadrics
166(4)
5.7 Bibliographical Note
170(1)
6 Chronological Calculus
171(20)
6.1 Motivation
171(1)
6.2 Duality
172(2)
6.2.1 On the Notation
174(1)
6.3 Topology on the Set of Smooth Functions
174(2)
6.3.1 Family of Functionals and Operators
175(1)
6.4 Operator ODEs and Volterra Expansion
176(6)
6.4.1 Volterra Expansion
177(3)
6.4.2 Adjoint Representation
180(2)
6.5 Variation Formulas
182(1)
6.6 Appendix: Estimates and Volterra Expansion
183(4)
6.7 Appendix: Remainder Term of the Volterra Expansion
187(3)
6.8 Bibliographical Note
190(1)
7 Lie Groups and Left-Invariant Sub-Riemannian Structures
191(22)
7.1 Subgroups of Diff(M) Generated by a Finite-Dimensional Lie Algebra of Vector Fields
191(7)
7.1.1 A Finite-Dimensional Approximation
193(3)
7.1.2 Passage to Infinite Dimension
196(1)
7.1.3 Proof of Proposition 7.2
197(1)
7.2 Lie Groups and Lie Algebras
198(10)
7.2.1 Lie Groups as Groups of Diffeomorphisms
200(2)
7.2.2 Matrix Lie Groups and Matrix Notation
202(3)
7.2.3 Bi-Invariant Pseudo-Metrics and Haar Measures
205(2)
7.2.4 The Levi-Malcev Decomposition
207(1)
7.3 Trivialization of TG and T*G
208(1)
7.4 Left-Invariant Sub-Riemannian Structures
209(2)
7.5 Example: Carnot Groups of Step
211(2)
210 7.5.1 Normal Pontryagin Extremals for Carnot Groups of Step 2
213(3)
7.6 Left-Invariant Hamiltonian Systems on Lie Groups
216(5)
7.6.1 Vertical Coordinates in TG and T*G
216(2)
7.6.2 Left-Invariant Hamiltonians
218(3)
7.7 Normal Extremals for Left-Invariant Sub-Riemannian Structures
221(11)
7.7.1 Explicit Expression for Normal Pontryagin Extremals in the s Case
221(2)
7.7.2 Example: The d s Problem on SO(3)
223(2)
7.7.3 Further Comments on the d s Problem: SO(3) and SO+ (2,1)
225(3)
7.7.4 Explicit Expression for Normal Pontryagin Extremals in the k z Case
228(4)
7.8 Rolling Spheres
232(12)
7.8.1 Rolling with Spinning
232(3)
7.8.2 Rolling without Spinning
235(5)
7.8.3 Euler's "Curvae Elasticae"
240(3)
7.8.4 Rolling Spheres: Further Comments
243(1)
7.9 Bibliographical Note
244(2)
8 Endpoint Map and Exponential Map
246(49)
8.1 The Endpoint Map
247(4)
8.1.1 Regularity of the Endpoint Map: Proof of Proposition 8.5
248(3)
8.2 Lagrange Multiplier Rule
251(1)
8.3 Pontryagin Extremals via Lagrange Multipliers
251(2)
8.4 Critical Points and Second-Order Conditions
253(8)
8.4.1 The Manifold of Lagrange Multipliers
256(5)
8.5 Sub-Riemannian Case
261(5)
8.6 Exponential Map and Gauss' lemma
266(4)
8.7 Conjugate Points
270(4)
8.8 Minimizing Properties of Extremal Trajectories
274(9)
8.8.1 Local Length-Minimality in the W1,2 Topology. Proof of Theorem 8.52
275(3)
8.8.2 Local Length-Minimality in the C° Topology
278(5)
8.9 Compactness of Length-Minimizers
283(2)
8.10 Cut Locus and Global Length-Minimizers
285(5)
8.11 An Example: First Conjugate Locus on a Perturbed Sphere
290(3)
8.12 Bibliographical Note
293(2)
9 2D Almost-Riemannian Structures
295(36)
9.1 Basic Definitions and Properties
295(11)
9.1.1 How Large is the Singular Set?
301(2)
9.1.2 Genuinely 2D Almost-Riemannian Structures Always Have Infinite Area
303(1)
9.1.3 Pontryagin Extremals
304(2)
9.2 The Grushin Plane
306(4)
9.2.1 Geodesies on the Grushin Plane
307(3)
9.3 Riemannian, Grushin and Martinet Points
310(5)
9.3.1 Normal Forms
313(2)
9.4 Generic 2D Almost-Riemannian Structures
315(3)
9.4.1 Proof of the Genericity Result
316(2)
9.5 A Gauss-Bonnet Theorem
318(11)
9.5.1 Integration of the Curvature
318(1)
9.5.2 The Euler Number
319(1)
9.5.3 Gauss-Bonnet Theorem
320(8)
9.5.4 Every Compact Orientable 2D Manifold can be Endowed with a Free Almost-Riemannian Structure with only Riemannian and Grushin Points
328(1)
9.6 Bibliographical Note
329(2)
10 Nonholonomic Tangent Space
331(45)
10.1 Flag of the Distribution and Carnot Groups
331(2)
10.2 Jet Spaces
333(5)
10.2.1 Jets of Curves
333(3)
10.2.2 Jets of Vector Fields
336(2)
10.3 Admissible Variations and Nonholonomic Tangent Space
338(5)
10.3.1 Admissible Variations
338(2)
10.3.2 Nonholonomic Tangent Space
340(3)
10.4 Nonholonomic Tangent Space and Privileged Coordinates
343(18)
10.4.1 Privileged Coordinates
343(3)
10.4.2 Description of the Nonholonomic Tangent Space in Privileged Coordinates
346(8)
10.4.3 Existence of Privileged Coordinates: Proof of Theorem 10.32
354(4)
10.4.4 Nonholonomic Tangent Spaces in Low Dimension
358(3)
10.5 Metric Meaning
361(6)
10.5.1 Convergence of the Sub-Riemannian Distance and the Ball-Box Theorem
362(5)
10.6 Algebraic Meaning
367(4)
10.6.1 Nonholonomic Tangent Space: The Equiregular Case
369(2)
10.7 Carnot Groups: Normal Forms in Low Dimension
371(4)
10.8 Bibliographical Note
375(1)
11 Regularity of the Sub-Riemannian Distance
376(26)
11.1 Regularity of the Sub-Riemannian Squared Distance
376(9)
11.2 Locally Lipschitz Functions and Maps
385(11)
11.2.1 Locally Lipschitz Map and Lipschitz Submanifolds
390(3)
11.2.2 A Non-Smooth Version of the Sard Lemma
393(3)
11.3 Regularity of Sub-Riemannian Spheres
396(3)
11.4 Geodesic Completeness and the Hopf-Rinow Theorem
399(1)
11.5 Bibliographical Note
400(2)
12 Abnormal Extremals and Second Variation
402(54)
12.1 Second Variation
402(2)
12.2 Abnormal Extremals and Regularity of the Distance
404(6)
12.3 Goh and Generalized Legendre Conditions
410(14)
12.3.1 Proof of Goh Condition-Part (i) of Theorem 12.12
413(7)
12.3.2 Proof of the Generalized Legendre Condition -- Part (ii) of Theorem 12.12
420(2)
12.3.3 More on the Goh and Generalized Legendre Conditions
422(2)
12.4 Rank-2 Distributions and Nice Abnormal Extremals
424(3)
12.5 Minimality of Nice Abnormal Extremals in Rank-2 Structures
427(9)
12.5.1 Proof of Theorem 12.33
429(7)
12.6 Conjugate Points along Abnormals
436(9)
12.6.1 Abnormals in Dimension 3
439(5)
12.6.2 Higher Dimensions
444(1)
12.7 Equivalence of Local Minimality with Respect to the W1,2 and C° Topologies
445(4)
12.8 Non-Minimality of Corners
449(5)
12.9 Bibliographical Note
454(2)
13 Some Model Spaces
456(57)
13.1 Carnot Groups of Step 2
457(3)
13.2 Multidimensional Heisenberg Groups
460(6)
13.2.1 Pontryagin Extremals in the Contact Case
461(2)
13.2.2 Optimal Synthesis
463(3)
13.3 Free Carnot Groups of Step 2
466(6)
13.3.1 Intersection of the Cut Locus with the Vertical Subspace
470(1)
13.3.2 The Cut Locus for the Free Step-2 Carnot Group of Rank 3
471(1)
13.4 An Extended Hadamard Technique to Compute the Cut Locus
472(6)
13.5 The Grushin Structure
478(8)
13.5.1 Optimal Synthesis Starting from a Riemannian Point
479(3)
13.5.2 Optimal Synthesis Starting from a Singular Point
482(4)
13.6 The Standard Sub-Riemannian Structure on SU(2)
486(4)
13.7 Optimal Synthesis on the Groups SO(3) and SO+(2,1)
490(4)
13.8 Synthesis for the Group of Euclidean Transformations of the Plane SE(2)
494(8)
13.8.1 Mechanical Interpretation
495(1)
13.8.2 Geodesies
496(6)
13.9 The Martinet Flat Sub-Riemannian Structure
502(7)
13.9.1 Abnormal Extremals
503(1)
13.9.2 Normal Extremals
504(5)
13.10 Bibliographical Note
509(4)
14 Curves in the Lagrange Grassmannian
513(29)
14.1 The Geometry of the Lagrange Grassmannian
513(6)
14.1.1 The Lagrange Grassmannian
517(2)
14.2 Regular Curves in the Lagrange Grassmannian
519(4)
14.3 Curvature of a Regular Curve
523(4)
14.4 Reduction of Non-Regular Curves in Lagrange Grassmannian
527(2)
14.5 Ample Curves
529(1)
14.6 From Ample to Regular
530(6)
14.7 Conjugate Points in L(E)
536(2)
14.8 Comparison Theorems for Regular Curves
538(2)
14.9 Bibliographical Note
540(2)
15 Jacobi Curves
542(9)
15.1 From Jacobi Fields to Jacobi Curves
542(3)
15.1.1 Jacobi Curves
543(2)
15.2 Conjugate Points and Optimality
545(2)
15.3 Reduction of the Jacobi Curves by Homogeneity
547(3)
15.4 Bibliographical Note
550(1)
16 Riemannian Curvature
551(20)
16.1 Ehresmann Connection
551(6)
16.1.1 Curvature of an Ehresmann Connection
552(2)
16.1.2 Linear Ehresmann Connections
554(1)
16.1.3 Covariant Derivative and Torsion for Linear Connections
555(2)
16.2 Riemannian Connection
557(6)
16.3 Relation to Hamiltonian Curvature
563(2)
16.4 Comparison Theorems for Conjugate Points
565(2)
16.5 Locally Flat Spaces
567(1)
16.6 Curvature of 2D Riemannian Manifolds
568(2)
16.7 Bibliographical Note
570(1)
17 Curvature in 3D Contact Sub-Riemannian Geometry
571(36)
17.1 A Worked-Out Example: The 2D Riemannian Case
571(5)
17.2 3D Contact Sub-Riemannian Manifolds
576(3)
17.3 Canonical Frames
579(3)
17.4 Curvature of a 3D Contact Structure
582(7)
17.4.1 Geometric Interpretation
588(1)
17.5 Local Classification of 3D Left-Invariant Structures
589(15)
17.5.1 A Description of the Classification
591(3)
17.5.2 A Sub-Riemannian Isometry Between Non-Isomorphic Lie groups
594(2)
17.5.3 Canonical Frames and Classification. Proof of Theorem 17.29
596(3)
17.5.4 An Explicit Isometry. Proof of Theorem 17.32
599(5)
17.6 Appendix: Remarks on Curvature Coefficients
604(1)
17.7 Bibliographical Note
605(2)
18 Integrability of the Sub-Riemannian Geodesic Flow on 3D Lie Groups
607(26)
18.1 Poisson Manifolds and Symplectic Leaves
607(5)
18.1.1 Poisson Manifolds
607(1)
18.1.2 The Poisson Bi-Vector
608(1)
18.1.3 Symplectic Manifolds
609(1)
18.1.4 Casimir Functions
609(2)
18.1.5 Symplectic Leaves
611(1)
18.2 Integrability of Hamiltonian Systems on Lie Groups
612(4)
18.2.1 The Poisson Manifold g*
612(2)
18.2.2 The Casimir First Integral
614(1)
18.2.3 First Integrals Associated with a Right-Invariant Vector Field
615(1)
18.2.4 Complete Integrability on Lie Groups
615(1)
18.3 Left-Invariant Hamiltonian Systems on 3D Lie Groups
616(16)
18.3.1 Rank-2 Sub-Riemannian Structures on 3D Lie Groups
621(2)
18.3.2 Classification of Symplectic Leaves on 3D Lie Groups
623(9)
18.4 Bibliographical Note
632(1)
19 Asymptotic Expansion of the 3D Contact Exponential Map
633(21)
19.1 The Exponential Map
633(3)
19.1.1 The Nilpotent Case
634(2)
19.2 General Case: Second-Order Asymptotic Expansion
636(5)
19.2.1 Proof of Proposition 19.2: Second-Order Asymptotics
637(4)
19.3 General Case: Higher-Order Asymptotic Expansion
641(12)
19.3.1 Proof of Theorem 19.6: Asymptotics of the Exponential Map
643(5)
19.3.2 Asymptotics of the Conjugate Locus
648(2)
19.3.3 Asymptotics of the Conjugate Length
650(1)
19.3.4 Stability of the Conjugate Locus
651(2)
19.4 Bibliographical Note
653(1)
20 Volumes in Sub-Riemannian Geometry
654(20)
20.1 Equiregular Sub-Riemannian Manifolds
654(1)
20.2 The Popp Volume
655(2)
20.3 A Formula for the Popp Volume in Terms of Adapted Frames
657(4)
20.4 The Popp Volume and Smooth Isometries
661(3)
20.5 Hausdorff Dimension and Hausdorff Volume
664(1)
20.6 Hausdorff Volume on Sub-Riemannian Manifolds
664(7)
20.6.1 Hausdorff Dimension
665(3)
20.6.2 On the Metric Convergence
668(1)
20.6.3 Induced Volumes and Estimates
669(2)
20.7 Density of the Spherical Hausdorff Volume with respect to a Smooth Volume
671(1)
20.8 Bibliographical Note
672(2)
21 The Sub-Riemannian Heat Equation
674(24)
21.1 The Heat Equation
674(10)
21.1.1 The Heat Equation in the Riemannian Context
674(4)
21.1.2 The Heat Equation in the Sub-Riemannian Context
678(3)
21.1.3 The Hormander Theorem and the Existence of the Heat Kernel
681(2)
21.1.4 The Heat Equation in the Non-Bracket-Generating Case
683(1)
21.2 The Heat Kernel on the Heisenberg Group
684(12)
21.2.1 The Heisenberg Group as a Group of Matrices
684(2)
21.2.2 The Heat Equation on the Heisenberg Group
686(1)
21.2.3 Construction of the Gaveau-Hulanicki Fundamental Solution
687(8)
21.2.4 Small-Time Asymptotics for the Gaveau-Hulanicki Fundamental Solution
695(1)
21.3 Bibliographical Note
696(2)
Appendix Geometry of Parametrized Curves in Lagrangian Grassmannians Igor Zelenko
698(27)
A.1 Preliminaries
698(6)
A.2 Algebraic Theory of Curves in Grassmannians and Flag Varieties
704(11)
A.3 Application to the Differential Geometry of Monotonic Parametrized Curves in Lagrangian Grassmannians
715(10)
References 725(15)
Index 740
Andrei Agrachev is currently a full professor at Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste. His research interests are: sub-Riemannian geometry, mathematical control theory, dynamical systems, differential geometry and topology, singularity theory and real algebraic geometry. Davide Barilari is Maītre de Conférence at Université de Paris VII (Denis Diderot). His research interests are: sub-Riemannian geometry, hypoelliptic operators, curvature and optimal transport. Ugo Boscain is Research Director at Centre National de la Recherche Scientifique (CNRS), Paris. His research interests are: sub-Riemannian geometry, hypoelliptic operators, quantum mechanics, singularity theory and geometric control.
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