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Curvature
The curvature discussed in this paper is a far reaching generalization of the Riemannian sectional curvature. The authors give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Rieman...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Providence :
American Mathematical Society,
2019.
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| Colección: | Memoirs of the American Mathematical Society Ser.
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| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 003 | OCoLC | ||
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| 008 | 190126s2019 riu o 000 0 eng d | ||
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| 020 | |a 9781470449131 | ||
| 020 | |a 1470449137 | ||
| 029 | 1 | |a AU@ |b 000069468753 | |
| 035 | |a (OCoLC)1083462573 | ||
| 050 | 4 | |a QA645 |b .A373 2018 | |
| 082 | 0 | 4 | |a 516.362 |2 23 |
| 049 | |a UAMI | ||
| 100 | 1 | |a Agrachev, A. | |
| 245 | 1 | 0 | |a Curvature |
| 260 | |a Providence : |b American Mathematical Society, |c 2019. | ||
| 300 | |a 1 online resource (154 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Memoirs of the American Mathematical Society Ser. ; |v v. 256 | |
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a Cover; Title page; Chapter 1. Introduction; 1.1. Structure of the paper; 1.2. Statements of the main theorems; 1.3. The Heisenberg group; Part 1 . Statements of the results; Chapter 2. General setting; 2.1. Affine control systems; 2.2. End-point map; 2.3. Lagrange multipliers rule; 2.4. Pontryagin Maximum Principle; 2.5. Regularity of the value function; Chapter 3. Flag and growth vector of an admissible curve; 3.1. Growth vector of an admissible curve; 3.2. Linearised control system and growth vector; 3.3. State-feedback invariance of the flag of an admissible curve | |
| 505 | 8 | |a 3.4. An alternative definitionChapter 4. Geodesic cost and its asymptotics; 4.1. Motivation: a Riemannian interlude; 4.2. Geodesic cost; 4.3. Hamiltonian inner product; 4.4. Asymptotics of the geodesic cost function and curvature; 4.5. Examples; Chapter 5. Sub-Riemannian geometry; 5.1. Basic definitions; 5.2. Existence of ample geodesics; 5.3. Reparametrization and homogeneity of the curvature operator; 5.4. Asymptotics of the sub-Laplacian of the geodesic cost; 5.5. Equiregular distributions; 5.6. Geodesic dimension and sub-Riemannian homotheties; 5.7. Heisenberg group | |
| 505 | 8 | |a 5.8. On the "meaning" of constant curvaturePart 2 . Technical tools and proofs; Chapter 6. Jacobi curves; 6.1. Curves in the Lagrange Grassmannian; 6.2. The Jacobi curve and the second differential of the geodesic cost; 6.3. The Jacobi curve and the Hamiltonian inner product; 6.4. Proof of Theorem; 6.5. Proof of Theorem; Chapter 7. Asymptotics of the Jacobi curve: Equiregular case; 7.1. The canonical frame; 7.2. Main result; 7.3. Proof of Theorem 7.4; 7.4. Proof of Theorem; 7.5. A worked out example: 3D contact sub-Riemannian structures; Chapter 8. Sub-Laplacian and Jacobi curves | |
| 505 | 8 | |a 8.1. Coordinate lift of a local frame8.2. Sub-Laplacian of the geodesic cost; 8.3. Proof of Theorem; Part 3 . Appendix; Appendix A. Smoothness of value function (Theorem 2.19); Appendix B. Convergence of approximating Hamiltonian systems (Proposition 5.15); Appendix C. Invariance of geodesic growth vector by dilations (Lemma 5.20); Appendix D. Regularity of (,) for the Heisenberg group (Proposition 5.51); Appendix E. Basics on curves in Grassmannians (Lemma 3.5 and 6.5); Appendix F. Normal conditions for the canonical frame | |
| 505 | 8 | |a Appendix G. Coordinate representation of flat, rank 1 Jacobi curves (Proposition 7.7)Appendix H.A binomial identity (Lemma 7.8); Appendix I.A geometrical interpretation of _{ }; Bibliography; Index; Back Cover | |
| 520 | |a The curvature discussed in this paper is a far reaching generalization of the Riemannian sectional curvature. The authors give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces. Special attention is paid to the sub-Riemannian (or Carnot-Carathéodory) metric spaces. The authors' construction of curvature is direct and naive, and similar to the original approach of Riemann. In particular, they extract geometric invariants from the asympto. | ||
| 590 | |a ProQuest Ebook Central |b Ebook Central Academic Complete | ||
| 650 | 0 | |a Curvature. | |
| 650 | 0 | |a Riemannian manifolds. | |
| 650 | 0 | |a Geometry, Differential. | |
| 650 | 6 | |a Courbure. | |
| 650 | 6 | |a Variétés de Riemann. | |
| 650 | 6 | |a Géométrie différentielle. | |
| 650 | 7 | |a Curvature |2 fast | |
| 650 | 7 | |a Geometry, Differential |2 fast | |
| 650 | 7 | |a Riemannian manifolds |2 fast | |
| 700 | 1 | |a Barilari, D. | |
| 700 | 1 | |a Rizzi, L. | |
| 758 | |i has work: |a Curvature (Text) |1 https://id.oclc.org/worldcat/entity/E39PCFHcCKphBcfPGDJrwvmTpP |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Agrachev, A. |t Curvature: a Variational Approach. |d Providence : American Mathematical Society, ©2019 |
| 830 | 0 | |a Memoirs of the American Mathematical Society Ser. | |
| 856 | 4 | 0 | |u https://ebookcentral.uam.elogim.com/lib/uam-ebooks/detail.action?docID=5633662 |z Texto completo |
| 938 | |a ProQuest Ebook Central |b EBLB |n EBL5633662 | ||
| 994 | |a 92 |b IZTAP | ||