Manfredo do Carmo (1928–2018)

Anyone knowing about Riemannian Geometry probably knows about this book. Do Carmo's Riemannian Geometry is one the most well-known books there is on the subject and, in my experience, it is recommended by pretty much everyone teaching it, and for good reasons. The book is advanced, of course, but it also tries to keep things simple, making everything easy to understand. Thanks of this, the book ends up being probably the best introduction the subject that one can ever get.

While a lot of show more important concepts are just sketched or omitted, like for example differential operators on manifolds, and covariant and Lie differentiation, these can be found on other well-known books like Petersen's Riemannian Geometry, which in my opinion however fails to be as clear as Do Carmo.

Definitely a solid textbook. Recommended to anyone who wants to dive into Riemannian Geometry just by knowing a bit of differential geometry. show less

LOCAL: SEDE DO IMPA
INTERVALO DE OCORRÊNCIA: 19 A 25 De Julho
N° DO COLÓQUIO: 16
IDIOMA: PORTUGUÊS

Esta versão consiste na tradução do seu livro publicado originalmente em inglês pela editora americana Prentice-Hall. O texto é um dos mais usados neste campo. Ele introduz à geometria diferencial de curvas e superfícies em seus aspectos local e global. A apresentação é feita com um uso mais extenso da álgebra linear elementar e com certo destaque para os fatos geométricos básicos em vez de cálculos mecânicos ou detalhes aleatórios.

O primeiro capítulo é dedicado às curvas. show more O segundo é desenvolvido com base no conceito de superfície regular em Rn, considerado pelo autor o melhor modelo, se bem apresentado, para o estudo das variedades diferenciáveis. Os demais capítulos mostram a aplicação normal de Gauss, a geometria intrínseca das superfícies em torno do conceito de derivada e trazem uma grande quantidade de geometria local das superfícies em R3. A edição é atualizada em relação as demais e vem com um conjunto de notas. show less

Indeholder "Preface", "Some Remarks on Using this Book", "1. Curves", " 1.1 Introduction", " 1.2 Parametrized Curves", " 1.3 Regular Curves; Arc Length", " 1.4 The Vector Product in R^3", " 1.5 The Local Theory of Curves Parametrized by Arc Length", " 1.6 The Local Canonical Form", " 1.7 Global Properties of Plane Curves", "2. Regular Surfaces", " 2.1 Introduction", " 2.2 Regular Surfaces; Inverse Images of Regular Values", " 2.3 Change of Parameters; Differential Functions on Surfaces", " show more 2.4 The Tangent Plane; the Differential of a Map", " 2.5 The First Fundamental Form; Area", " 2.6 Orientation of Surfaces", " 2.7 A Characterization of Compact Orientable Surfaces", " 2.8 A Geometric Definition of Area", " Appendix: A Brief Review on Continuity and Differentiability", "3. The Geometry of the Gauss Map", " 3.1 Introduction", " 3.2 The Definition of the Gauss Map and Its Fundamental Properties", " 3.3 The Gauss Map in Local Coordinates", " 3.4 Vector Fields", " 3.5 Ruled Surfaces and Minimal Surfaces", " Appendix: Self-Adjoint Linear Maps and Quadratic Forms", "4. The Intrinsic Geometry of Surfaces", " 4.1 Introduction", " 4.2 Isometries; Conformal Maps", " 4.3 The Gauss Theorem and the Equations of Compatibility", " 4.4 Parallel Transport; Geodesics", " 4.5 The Gauss-Bonnet Theorem and its Applications", " 4.6 The Exponential Map. Geodesic Polar Coordinates", " 4.7 Further Properties of Geodesics. Convex Neighborhoods", " Appendix: Proofs of the Fundamental Theorems of The Local Theory of Curves and Surfaces", "5. Global Differential Geometry", " 5.1 Introduction", " 5.2 The Rigidity of the Sphere", " 5.3 Complete Surfaces. Theorem of Hopf-Rinow", " 5.4 First and Second Variations of the Arc Length; Bonnet's Theorem", " 5.5 Jacobi Fields and Conjugate Points", " 5.6 Covering Spaces; the Theorems of Hadamard", " 5.7 Global Theorems for Curves; the Fary-Milnor Theorem", " 5.8 Surfaces of Zero Gaussian Curvature", " 5.9 Jacobi's Theorems", " 5.10 Abstract Surfaces; Further Generalizations", " 5.11 Hilbert's Theorem", " Appendix: Point-Set Topology of Euclidean Spaces", "Bibliography and Comments", "Hints and Answers to Some Exercises", "Index".

Standardlærebog i differentialgeometri show less