Riemannian geometry / Manfredo Perdigão do Carmo ; translated by Francis Flaherty.
Idioma: Inglés Lenguaje original: Portugués Series Mathematics (Boston, Mass.): Editor: Boston : Birkhäuser, c1992Descripción: 300 p. : il. ; 25 cmISBN: 0817634908; 3764334908Títulos uniformes: Geometria riemanniana. Inglés Tema(s): Geometry, RiemannianOtra clasificación: 53-01 Recursos en línea: Publisher descriptionPreface to the first edition ix Preface to the second edition x Preface to the English edition xi How to use this book xii CHAPTER 0—DIFFERENTIABLE MANIFOLDS [1] §1. Introduction [1] §2. Differentiable manifolds; tangent space [2] §3. Immersions and embeddings; examples [11] §4. Other examples of manifolds. Orientation [15] §5. Vector fields; brackets. Topology of manifolds [25] CHAPTER 1—RIEMANNIAN METRICS [35] §1. Introduction [35] §2. Riemannian Metrics [38] CHAPTER 2—AFFINE CONNECTIONS; RIEMANNIAN CONNECTIONS [48] §1. Introduction [48] §2. Affine connections [49] §3. Riemannian connections [53] CHAPTER 3—GEODESICS; CONVEX NEIGHBORHOODS [60] §1. Introduction [60] §2. The geodesic flow [61] §3. Minimizing properties of geodesics [67] §4. Convex neighborhoods [75] CHAPTER 4—CURVATURE [88] §1. Introduction [88] §2. Curvature [89] §3. Sectional curvature [93] §4. Ricci curvature and scalar curvature [97] §5. Tensors on Riemannian manifolds [100] CHAPTER 5—JACOBI FIELDS [110] §1. Introduction [110] §2. The Jacobi equation [110] §3. Conjugate points [116] CHAPTER 6—ISOMETRIC IMMERSIONS [124] §1. Introduction [124] §2. The second fundamental form [125] §3. The fundamental equations [134] CHAPTER 7—COMPLETE MANIFOLDS; HOPF-RINOW AND HADAMARD THEOREMS [144] §1. Introduction [144] §2. Complete manifolds; Hopf-Rinow Theorem [145] §3. The Theorem of Hadamard [149] CHAPTER 8—SPACES OF CONSTANT CURVATURE [155] §1. Introduction [155] §2. Theorem of Cartan on the determination of the metric by means of the curvature [156] §3. Hyperbolic space [160] §4. Space forms [162] §5. Isometries of the hyperbolic space; Theorem of Liouville [168] CHAPTER 9-VARIATIONS OF ENERGY [191] §1. Introduction [191] §2. Formulas for the first and second variations of energy [191] §3. The theorems of Bonnet-Myers and of Synge-Weinstein [200] CHAPTER 10—THE RAUCH COMPARISON THEOREM [210] §1. Introduction [210] §2. The Theorem of Rauch [212] §3. Applications of the Index Lemma to immersions [221] §4. Focal points and an extension of Rauch’s Theorem [227] CHAPTER 11—THE MORSE INDEX THEOREM [242] §1. Introduction [242] §2. The Index Theorem [242] CHAPTER 12—THE FUNDAMENTAL GROUP OF MANIFOLDS OF NEGATIVE CURVATURE [253] §1. Introduction [253] §2. Existence of closed geodesics [254] §3. Preissman’s Theorem [258] CHAPTER 13—THE SPHERE THEOREM [265] §1. Introduction [265] §2. The cut locus [267] §3. The estimate of the injectivity radius [276] §4. The Sphere Theorem [283] §5. Some further developments [288] References [292] Index [297]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 53 C287ri (Browse shelf) | Available | A-6812 |
Browsing Instituto de Matemática, CONICET-UNS shelves, Shelving location: Libros ordenados por tema Close shelf browser
| 53 C287 Differential geometry of curves and surfaces / | 53 C287m O método do referencial móvel / | 53 C287r Geometria riemanniana / | 53 C287ri Riemannian geometry / | 53 C322-2 Leçons sur la géométrie des espaces de Riemann / | 53 C322f Les espaces de Finsler / | 53 C322m Les espaces métriques : |
Traducción de: Geometria riemanniana. 2.ª ed.
Incluye referencias bibliográficas (p. [292]-296) e índice.
Preface to the first edition ix --
Preface to the second edition x --
Preface to the English edition xi --
How to use this book xii --
CHAPTER 0—DIFFERENTIABLE MANIFOLDS [1] --
§1. Introduction [1] --
§2. Differentiable manifolds; tangent space [2] --
§3. Immersions and embeddings; examples [11] --
§4. Other examples of manifolds. Orientation [15] --
§5. Vector fields; brackets. Topology of manifolds [25] --
CHAPTER 1—RIEMANNIAN METRICS [35] --
§1. Introduction [35] --
§2. Riemannian Metrics [38] --
CHAPTER 2—AFFINE CONNECTIONS; --
RIEMANNIAN CONNECTIONS [48] --
§1. Introduction [48] --
§2. Affine connections [49] --
§3. Riemannian connections [53] --
CHAPTER 3—GEODESICS; CONVEX NEIGHBORHOODS [60] --
§1. Introduction [60] --
§2. The geodesic flow [61] --
§3. Minimizing properties of geodesics [67] --
§4. Convex neighborhoods [75] --
CHAPTER 4—CURVATURE [88] --
§1. Introduction [88] --
§2. Curvature [89] --
§3. Sectional curvature [93] --
§4. Ricci curvature and scalar curvature [97] --
§5. Tensors on Riemannian manifolds [100] --
CHAPTER 5—JACOBI FIELDS [110] --
§1. Introduction [110] --
§2. The Jacobi equation [110] --
§3. Conjugate points [116] --
CHAPTER 6—ISOMETRIC IMMERSIONS [124] --
§1. Introduction [124] --
§2. The second fundamental form [125] --
§3. The fundamental equations [134] --
CHAPTER 7—COMPLETE MANIFOLDS; HOPF-RINOW AND HADAMARD THEOREMS [144] --
§1. Introduction [144] --
§2. Complete manifolds; Hopf-Rinow Theorem [145] --
§3. The Theorem of Hadamard [149] --
CHAPTER 8—SPACES OF CONSTANT CURVATURE [155] --
§1. Introduction [155] --
§2. Theorem of Cartan on the determination of the metric by means of the curvature [156] --
§3. Hyperbolic space [160] --
§4. Space forms [162] --
§5. Isometries of the hyperbolic space; Theorem of Liouville [168] --
CHAPTER 9-VARIATIONS OF ENERGY [191] --
§1. Introduction [191] --
§2. Formulas for the first and second variations of energy [191] --
§3. The theorems of Bonnet-Myers and of Synge-Weinstein [200] --
CHAPTER 10—THE RAUCH COMPARISON THEOREM [210] --
§1. Introduction [210] --
§2. The Theorem of Rauch [212] --
§3. Applications of the Index Lemma to immersions [221] --
§4. Focal points and an extension of Rauch’s Theorem [227] --
CHAPTER 11—THE MORSE INDEX THEOREM [242] --
§1. Introduction [242] --
§2. The Index Theorem [242] --
CHAPTER 12—THE FUNDAMENTAL GROUP OF MANIFOLDS OF NEGATIVE CURVATURE [253] --
§1. Introduction [253] --
§2. Existence of closed geodesics [254] --
§3. Preissman’s Theorem [258] --
CHAPTER 13—THE SPHERE THEOREM [265] --
§1. Introduction [265] --
§2. The cut locus [267] --
§3. The estimate of the injectivity radius [276] --
§4. The Sphere Theorem [283] --
§5. Some further developments [288] --
References [292] --
Index [297] --
MR, 92i:53001
Click on an image to view it in the image viewer
There are no comments on this title.