微分幾何中嘉當的活動標架法和外微分係統初步（英文版） [Cartan for Beginners:Differential Geometry Via Moving Frames and Exterio 下載 mobi epub pdf 電子書 2024

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　　《微分幾何中嘉當的活動標架法和外微分係統初步（英文版）》介紹瞭微分幾何的嘉當方法。嘉當幾何的兩個中心方法是外微分理論和活動標架方法，《微分幾何中嘉當的活動標架法和外微分係統初步（英文版）》對它們做瞭深入和現代化的處理，包括它們在古典和現代問題中的應用。
　　《微分幾何中嘉當的活動標架法和外微分係統初步（英文版）》一開始用活動標架的語言講述瞭經典麯麵幾何和基礎黎曼幾何，然後簡要介紹瞭外微分。很多關鍵概念是通過導嚮定義、定理和證明的有啓發性的例子逐步展開的。
　　這些方法的基礎建立後，作者便轉嚮應用和更高深的專題。一個引人注目的應用是關於復代數幾何的，在那裏射影微分幾何的一些重要結果得以拓展和更新。本書重點引進瞭G-結構並討論瞭聯絡理論。通過Darboux方法、特徵法、等價性的嘉當法，嘉當的這種機製也被用來求偏微分方程的顯式解。
　　《微分幾何中嘉當的活動標架法和外微分係統初步（英文版）》適閤在一年期的微分幾何研究生課程中講授，通篇包括大量的習題和例題。偏微分方程和代數幾何等方嚮的專傢如果想瞭解活動標架和外微分在他們領域中的應用，那麼本書對他們是十分有用的。

內頁插圖

目錄

Preface
Chapter 1． Moving Frames and Exterior Differential Systems
1．1． Geometry of surfaces in E3 in coordinates
1．2． Differential equations in coordinates
1．3． Introduction to differential equations without coordinates
1．4． Introduction to geometry without coordinates： curves in E2
1．5． Submanifolds of homogeneous spaces
1．6． The Maurer-Cartan form
1．7． Plane curves in other geometries
1．8． Curves in E3
1．9． Exterior differential systems and jet spaces

Chapter 2． Euclidean Geometry and Riemannian Geometry
2．1． Gauss and mean curvature via frames
2．2． Calculation of H and K for some examples
2．3． Darboux frames and applications
2．4． What do H and K tell us？
2．5． Invariants for n，-dimensional submanifolds of En+s
2．6． Intrinsic and extrinsic geometry
2．7． Space forms： the sphere and hyperbolic space
2．8． Curves on surfaces
2．9． The Gauss-Bonnet and Poincare-Hopf theorems
2．10． Non-orthonormalframes

Chapter 3． Projective Geometry
3．1． Grassmannians
3．2． Frames and the projective second fundamental form
3．3． Algebraic varieties
3．4． Varieties with degenerate Gauss mappings
3．5． Higher-order differential invariants
3．6． Fundamental forms of some homogeneous varieties
3．7． Higher-order Fubini forms
3．8． Ruled and uniruled varieties
3．9． Varieties with vanishing Fubini cubic
3．10． Dual varieties
3．11． Associated varieties
3．12． More on varieties with degenerate Gauss maps
3．13． Secant and tangential varieties
3．14． Rank restriction theorems
3．15． Local study of smooth varieties with degenerate tangential varieties
3．16． Generalized Monge systems
3．17． Complete intersections

Chapter 4． Cartan-Kahler Ⅰ： Linear Algebra and Constant-Coefficient Homogeneous Systems
4．1． Tableaux
4．2． First example
4．3． Second example
4．4． Third example
4．5． The general case
4．6． The characteristic variety of a tableau

Chapter 5． Cartan-Kahler Ⅱ： The Cartan Algorithm for Linear Pfaffian Systems
5．1． Linear Pfaffian systems
5．2． First example
5．3． Second example： constant coefficient homogeneous systems
5．4． The local isometric embedding problem
5．5． The Cartan algorithm formalized： tableau， torsion and prolongation
5．6． Summary of Cartan's algorithm for linear Pfaffian systems
5．7． Additional remarks on the theory
5．8． Examples
5．9． Functions whose Hessians commute， with remarks on singular solutions
5．10． The Cartan-Janet Isometric Embedding Theorem
5．11． Isometric embeddings of space forms （mostly flat ones）
5．12． Calibrated submanifolds

Chapter 6． Applications to PDE
6．1． Symmetries and Cauchy characteristics
6．2． Second-order PDE and Monge characteristics
6．3． Derived systems and the method of Darboux
6．4． Monge-Ampere systems and Weingarten surfaces
6．5． Integrable extensions and Backlund transformations

Chapter 7． Cartan-Kahler Ⅲ： The General Case
7．1． Integral elements and polar spaces
7．2． Example： Triply orthogonal systems
7．3． Statement and proof of Cartan-Kahler
7．4． Cartan's Test
7．5． More examples of Cartan's Test

Chapter 8． Geometric Structures and Connections
8．1． G-structures
8．2． How to differentiate sections of vector bundles
8．3． Connections on Fc and differential invariants of G-structures
8．4． Induced vector bundles and connections on induced bundles
8．5． Holonomy
8．6． Extended example： Path geometry
8．7． Frobenius and generalized conformal structures

Appendix A． Linear Algebra and Representation Theory
A．1． Dual spaces and tensor products
A．2． Matrix Lie groups
A．3． Complex vector spaces and complex structures
A．4． Lie algebras
A．5． Division algebras and the simple group G2
A．6． A smidgen of representation theory
A．7． Clifford algebras and spin groups

Appendix B． Differential Forms
B．1． Differential forms and vector fields
B．2． Three definitions of the exterior derivative
B．3． Basic and semi-basic forms
B．4． Differentialideals

Appendix C． Complex Structures and Complex Manifolds
C．1． Complex manifolds
C．2． The Cauchy-Riemann cquations
Appendix D． Initial Value Problems
Hints and Answers to Selected Exercises
Bibliography
Index

前言/序言

　　In this book， we use moving frames and exterior differential systems to study geometry and partial differential equations. These ideas originated about a century ago in the works of several mathematicians， including Gaston Darboux， Edouard Goursat and， most importantly， Elie Cartan. Over the years these techniques have been refined and extended; major contributors to the subject are mentioned below， under "Further Reading".
　　The book has the following features： It concisely covers the classical geometry of surfaces and basic Riemannian geometry in the language of moving frames. It includes results from projective differential geometry that update and expand the classic paper [69] of Griffiths and Harris. It provides an elementary introduction to the machinery of exterior differential systems (EDS)， and an introduction to the basics of G-structures and the general theory of connections. Classical and recent geometric applications of these techniques are discussed throughout the text.
　　This book is intended to be used as a textbook for a graduate-level course; there are numerous exercises throughout. It is suitable for a one- year course， although it has more material than can be covered in a year， and parts of it are suitable for one-semester course (see the end of this preface for some suggestions). The intended audience is both graduate students who have some familiarity with classical differential geometry and differentiable manifolds， and experts in areas such as PDE and algebraic geometry who want to learn how moving frame and EDS techniques apply to their fields.
　　In addition to the geometric applications presented here， EDS techniques are also applied in CR geometry (see， e.g.， [98])， robotics， and control theory (see [55， 56， 129]). This book prepares the reader for such areas， as well as for more advanced texts on exterior differential systems， such as [20]， and papers on recent advances in the theory， such as [58， 117l. Overview. Each section begins with geometric examples and problems. Techniques and definitions are introduced when they become useful to help solve the geometric questions under discussion. We generally keep the pre- sentation elementary， although advanced topics are interspersed throughout the text.
　　In Chapter 1， we introduce moving frames via the geometry of curves in the Euclidean plane E2. We define the Maurer-Cartan form of a Lie group G and explain its use in the study of submanifolds of G-homogeneous spaces. We give additional examples， including the equivalence of holomorphic mappings up to fractional linear transformation， where the machinery leads one naturally to the Schwarzian derivative.
　　We define exterior differential systems and jet spaces， and explain how to rephrase any system of partial differential equations as an EDS using jets. We state and prove the Frobenius system， leading up to it via an elementary example of an overdetermined system of PDE.
　　In Chapter 2， we cover traditional material-the geometry of surfaces in three-dimensional Euclidean space， submanifolds of higher-dimensional Eu- clidean space， and the rudiments of Riemannian geometry-all using moving frames. Our emphasis is on local geometry， although we include standard global theorems such as the rigidity of the sphere and the Gauss-Bonnet Theorem. Our presentation emphasizes finding and interpreting differential invariants to enable the reader to use the same techniques in other settings.
　　We begin Chapter 3 with a discussion of Grassmannians and the Pliicker embedding. We present some well-known material (e.g.， Fubini's theorem on the rigidity of the quadric) which is not readily available in other textbooks. We present several recent results， including the Zak and Landman theorems on the dual defect， and results of the second author on complete intersections. osculating hypersurface 微分幾何中嘉當的活動標架法和外微分係統初步（英文版） [Cartan for Beginners:Differential Geometry Via Moving Frames and Exterio 下載 mobi epub pdf txt 電子書 格式

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