![]() ![]() A special feature of the book is that it deals with infinite-dimensional manifolds, modeled on a Banach space in general, and a Hilbert space for Riemannian geometry. "The text provides a valuable introduction to basic concepts and fundamental results in differential geometry. It can be warmly recommended to a wide audience." "There are many books on the fundamentals of differential geometry, but this one is quite exceptional this is not surprising for those who know Serge Lang's books. A certain number of concepts are essential for all three, and are so basic and elementary that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginnings. In differential equations, one studies vector fields and their in tegral curves, singular points, stable and unstable manifolds, etc. Formally, one may say that one studies properties invariant under the group of differentiable automorphisms which preserve the additional structure. ) and studies properties connected especially with these objects. In differential geometry, one puts an additional structure on the differentiable manifold (a vector field, a spray, a 2-form, a Riemannian metric, ad lib. One may also use differentiable structures on topological manifolds to deter mine the topological structure of the manifold (for example, it la Smale ). In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differen tiable maps in them (immersions, embeddings, isomorphisms, etc. At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. The size of the book influenced where to stop, and there would be enough material for a second volume (this is not a threat). Lee's: Manifolds and Differential Geometry.The present book aims to give a fairly comprehensive account of the fundamentals of differential manifolds and differential geometry. Once you get a foundation in Differential Geometry, I'd recommend looking into J.Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout." "Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. ![]() Also look into the book with the same title: Elementary Differential Geometry, 2nd Ed (2010),, this one authored by Andrew Pressley."Written primarily for students who have completed the standard first courses in calculus and linear algebra, it provides an introduction to the geometry of curves and surfaces." You might also want to look into O'Neill's Elementary Differential Geometry, perhaps a good choice to start off with.Other choices for Elementary Introductions: There are also lecture notes that accompany the course and text. Providence, RI: American Mathematical Society, 2002. Differential Geometry: Curves – Surfaces – Manifolds. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature. This course is an introduction to differential geometry. You might want to check out the the course on Differential Geometry via MIT Open Course Ware, (Prof. Each is also accompanied by credible "reviews", which may help you select the appropriate text(s) to meet your needs.Īs you seem to be looking for a more elementary introduction to differential geometry: Riemannian Manifolds: An Introduction to CurvatureĮach of the above links to Amazon, simply because you can preview the texts, e.g., the Table(s) of Contents, to see if any/all meet your needs.There's no arguing that John Lee's texts are excellent: the following are part of the series " Graduate Texts in Mathematics":
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