Curvature: a variational approach:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2018]
|
| Series: | Memoirs of the American Mathematical Society
volume 256, number 1225 (first of 6 numbers) |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | v, 142, 4 ungezählte Seiten Illustrationen, Diagramme |
| ISBN: | 9781470426460 |
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MARC
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| 100 | 1 | |a Agračev, Andrej Aleksandrovič |d 1952- |0 (DE-588)1089340044 |4 aut | |
| 245 | 1 | 0 | |a Curvature: a variational approach |c A. Agrachev, D. Barilari, L. Rizzi |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2018] | |
| 264 | 4 | |c © 2018 | |
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Record in the Search Index
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| adam_text | Contents
Chapter 1. Introduction 1
1.1. Structure of the paper 5
1.2. Statements of the main theorems 6
1.3. The Heisenberg group 7
Part 1. Statements of the results 9
Chapter 2. General setting 11
2.1. Affine control systems 11
2.2. End-point map 14
2.3. Lagrange multipliers rule 15
2.4. Pontryagin Maximum Principle 16
2.5. Regularity of the value function 17
Chapter 3. Flag and growth vector of an admissible curve 19
3.1. Growth vector of an admissible curve 19
3.2. Linearised control system and growth vector 21
3.3. State-feedback invariance of the flag of an admissible curve 24
3.4. An alternative definition 25
Chapter 4. Geodesic cost and its asymptotics 29
4.1. Motivation: a Riemannian interlude 29
4.2. Geodesic cost 30
4.3. Hamiltonian inner product 31
4.4. Asymptotics of the geodesic cost function and curvature 32
4.5. Examples 35
Chapter 5. Sub-Riemannian geometry 41
5.1. Basic definitions 41
5.2. Existence of ample geodesics 46
5.3. Reparametrization and homogeneity of the curvature operator 49
5.4. Asymptotics of the sub-Laplacian of the geodesic cost 50
5.5. Equiregular distributions 53
5.6. Geodesic dimension and sub-Riemannian homotheties 56
5.7. Heisenberg group 58
5.8. On the “meaning” of constant curvature 63
Part 2. Technical tools and proofs 67
Chapter 6. Jacobi curves 69
6.1. Curves in the Lagrange Grassmannian 69
iii
iv CONTENTS
6.2. The Jacobi curve and the second differential of the geodesic cost 71
6.3. The Jacobi curve and the Hamiltonian inner product 75
6.4. Proof of Theorem A 76
6.5. Proof of Theorem D 80
Chapter 7. Asymptotics of the Jacobi curve: Equiregular case 83
7.1. The canonical frame 83
7.2. Main result 86
7.3. Proof of Theorem 7.4 88
7.4. Proof of Theorem B 92
7.5. A worked out example: 3D contact sub-Riemannian structures 93
Chapter 8. Sub-Laplacian and Jacobi curves 107
8.1. Coordinate lift of a local frame 108
8.2. Sub-Laplacian of the geodesic cost 108
8.3. Proof of Theorem C 109
Part 3. Appendix 113
Appendix A. Smoothness of value function (Theorem 2.19) 115
Appendix B. Convergence of approximating Hamiltonian systems
(Proposition 5.15) 119
Appendix C. Invariance of geodesic growth vector by dilations (Lemma 5.20) 121
Appendix D. Regularity of C(t, s) for the Heisenberg group (Proposition 5.51) 123
Appendix E. Basics on curves in Grassmannians (Lemma 3.5 and 6.5) 125
Appendix F. Normal conditions for the canonical frame 127
Appendix G. Coordinate representation of flat, rank 1 Jacobi curves
(Proposition 7.7) 129
Appendix H. A binomial identity (Lemma 7.8) 131
Appendix I. A geometrical interpretation of c* 135
Bibliography 137
Index
141
|
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| illustrated | Illustrated |
| indexdate | 2024-07-10T08:18:59Z |
| institution | BVB |
| isbn | 9781470426460 |
| language | English |
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| physical | v, 142, 4 ungezählte Seiten Illustrationen, Diagramme |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Memoirs of the American Mathematical Society |
| series2 | Memoirs of the American Mathematical Society |
| spelling | Agračev, Andrej Aleksandrovič 1952- (DE-588)1089340044 aut Curvature: a variational approach A. Agrachev, D. Barilari, L. Rizzi Providence, Rhode Island American Mathematical Society [2018] © 2018 v, 142, 4 ungezählte Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Memoirs of the American Mathematical Society volume 256, number 1225 (first of 6 numbers) Krümmung (DE-588)4128765-4 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Krümmung (DE-588)4128765-4 s Riemannsche Geometrie (DE-588)4128462-8 s DE-604 Barilari, Davide 1984- (DE-588)112004104X aut Rizzi, L. aut Memoirs of the American Mathematical Society volume 256, number 1225 (first of 6 numbers) (DE-604)BV008000141 1225 V:DE-605;X:Imageware application/pdf http://digitale-objekte.hbz-nrw.de/storage2/2019/02/03/file_21/8251722.pdf Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030853224&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Agračev, Andrej Aleksandrovič 1952- Barilari, Davide 1984- Rizzi, L. Curvature: a variational approach Memoirs of the American Mathematical Society Krümmung (DE-588)4128765-4 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)4128765-4 (DE-588)4128462-8 |
| title | Curvature: a variational approach |
| title_auth | Curvature: a variational approach |
| title_exact_search | Curvature: a variational approach |
| title_full | Curvature: a variational approach A. Agrachev, D. Barilari, L. Rizzi |
| title_fullStr | Curvature: a variational approach A. Agrachev, D. Barilari, L. Rizzi |
| title_full_unstemmed | Curvature: a variational approach A. Agrachev, D. Barilari, L. Rizzi |
| title_short | Curvature: a variational approach |
| title_sort | curvature a variational approach |
| topic | Krümmung (DE-588)4128765-4 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Krümmung Riemannsche Geometrie |
| url | http://digitale-objekte.hbz-nrw.de/storage2/2019/02/03/file_21/8251722.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030853224&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008000141 |
| work_keys_str_mv | AT agracevandrejaleksandrovic curvatureavariationalapproach AT barilaridavide curvatureavariationalapproach AT rizzil curvatureavariationalapproach |
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