Projective Differential Geometry Old and New pdf epub mobi txt 电子书 下载 2026

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出版者:Cambridge University Press

作者:V. Ovsienko

出品人:

页数:262

译者:

出版时间:2004-12-13

价格:USD 119.00

装帧:Hardcover

isbn号码:9780521831864

丛书系列:

图书标签:

• 数学
• 微分几何
• 射影几何
• 几何学
• 数学
• 高等数学
• 拓扑学
• 代数几何
• 经典数学
• 现代数学
• 数学史

Projective Differential Geometry: A Bridge Between Geometry and Analysis This book offers a comprehensive exploration of projective differential geometry, a field that gracefully intertwines the intrinsic properties of curves and surfaces with the powerful tools of calculus and differential equations. It delves into how projective transformations, which preserve lines and ratios of distances along lines, reveal fundamental geometric invariants that are otherwise obscured by Euclidean measurements. Part I: Foundations in Projective Space The journey begins with a rigorous introduction to projective space. We will establish the fundamental concepts of homogeneous coordinates, lines, planes, and higher-dimensional projective spaces. The geometric interpretation of projective transformations, including their matrices and canonical forms, will be meticulously laid out. Special attention will be paid to the concept of polarity and its profound implications in relating points and lines, as well as conics and quadrics. This foundational section will equip the reader with the essential language and framework for understanding the subsequent geometric developments. Part II: Projective Properties of Curves The focus then shifts to curves, where we will investigate their behavior under projective transformations. The concept of the projective tangent and osculating plane will be introduced, and we will derive differential invariants that characterize curves projectively. This will involve exploring the fundamental theorems of projective curve theory, such as the classification of projectively equivalent curves. The relationship between curvature in Euclidean geometry and its projective counterparts will be examined, highlighting how projective invariants offer a more intrinsic and general description. We will also delve into the study of special curves with significant projective properties. Part III: Projective Geometry of Surfaces Building upon the theory of curves, we will extend our investigations to surfaces. The projective normal, geodesic, and asymptotic lines will be defined, and their projective significance will be elucidated. The core of this section lies in the development of projective surface theory, which aims to classify surfaces based on their behavior under projective transformations. We will introduce and meticulously analyze the fundamental equations of projective differential geometry for surfaces, including the determination of projective curvature. Key theorems, such as the Dupin cyclide in a projective context, will be explored, revealing the deep connections between differential invariants and projective equivalence. The section will also touch upon special types of surfaces possessing unique projective characteristics. Part IV: Advanced Topics and Modern Connections The latter part of the book ventures into more advanced and contemporary aspects of projective differential geometry. This will include exploring connections with affine differential geometry, highlighting the transition from projective to affine invariants. We will investigate the role of Lie groups and their actions in understanding projective transformations and their invariants. The geometric interpretation of certain differential equations within the projective framework will be examined, showcasing the interplay between geometry and analysis. Furthermore, the book will touch upon modern research directions and potential applications, such as connections to algebraic geometry and areas of theoretical physics where projective structures play a crucial role. Throughout the book, the emphasis will be on a clear and intuitive geometric understanding, supported by rigorous analytical derivations. The interplay between visual intuition and formal mathematical proof will be a guiding principle. While no prior knowledge of projective geometry is strictly assumed, a solid understanding of differential calculus and basic linear algebra will be beneficial. The material is structured to be accessible to graduate students and researchers in mathematics and related fields seeking to deepen their understanding of geometric structures and their transformations. The "Old and New" in the title reflects both the established classical results and the modern reinterpretations and extensions that continue to make this field vibrant and relevant.

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这本书的习题设置是其教学价值的另一个亮点，但同时也是一个不小的挑战。这些练习题显然不是那种简单的代入公式就能得出答案的“水题”。它们的设计目的似乎是为了巩固对核心思想的掌握，而不是仅仅测试计算能力。许多题目都要求读者进行深入的构造性证明，或者需要读者自己去“发现”隐藏在理论结构之下的某种对称性或不变性。我感觉，如果能扎扎实实地完成其中的大部分题目，那么对这门学科的理解深度将得到质的飞跃。不过，我也必须坦诚，对于基础薄弱的读者而言，这些习题可能会显得有些“劝退”，它们无疑是为那些已经准备好迎接智力挑战的学习者准备的，是检验学习成果的试金石，而非简单的课后作业。

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从内容的广度来看，这本书似乎在试图构建一个连接经典与现代的桥梁。我能感受到作者对于不同流派思想的融合抱有极大的热情，它不满足于仅仅停留在某个特定分支的深入挖掘，而是努力勾勒出整个学科领域的全景图。这对于那些希望跳出单一研究领域，寻求跨学科理解的读者来说，具有不可估量的价值。书中对某些概念的定义和解释，明显带有作者个人的深刻印记和独特的视角，这使得它区别于那些千篇一律的、由多个编委会共同撰写的参考书。这种“个性化”的论述，虽然可能需要读者具备一定的背景知识来更好地消化，但一旦融会贯通，带来的启发性是巨大的，它激发了读者主动去探索更多相关领域的好奇心。

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虽然我还没有来得及深入研读每一个定理的证明细节，但仅从目录的梳理和章节间的逻辑推进来看，这本书的整体结构呈现出一种极强的系统性。它似乎并没有急于展示那些最尖端、最晦涩的成果，而是采取了一种“追本溯源”的教学策略。开篇对基础概念的定义和铺垫非常扎实，像是为初学者搭建了一个坚固的地基，确保读者在进入更复杂的几何结构之前，对核心思想有着深刻的理解。我注意到作者似乎很注重历史脉络的梳理，许多现代观点都是在回顾经典理论的局限性后自然引出的，这种叙事方式极大地降低了抽象概念的理解门槛。它不像某些教科书那样，上来就扔一堆公理让你硬背，而是通过循序渐进的方式，让读者“理解”为什么需要引入这些新的几何框架，这种教学上的匠心独运，使得学习过程变得更为流畅和有意义。

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这本书的语言风格倒是颇有些令人玩味。它不像某些翻译作品那样，充斥着生硬的学术腔调，读起来像是在啃一块味同嚼蜡的干面包。相反，作者的笔触显得相当克制而精确，用词选择非常讲究，既保持了数学描述的严谨性，又在一些关键的过渡和解释部分，流露出一种老派绅士般的优雅和从容。我尤其欣赏它在引入新概念时，常常伴随着一些富有洞察力的比喻或者类比，这些“拐点”性的描述，往往能瞬间点亮原本模糊不清的思维角落。当然，这并不意味着它会过度口语化，毕竟这是一本严肃的数学著作，但正是这种平衡感，使得长时间的阅读也未曾感到心神俱疲。它像是与一位饱学之士在安静的图书馆里进行一场深入的探讨，而不是被动地接受信息灌输。

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这本书的装帧设计着实令人眼前一亮，封面那种低饱和度的深蓝搭配烫金的标题，散发着一种低调的学术气息。拿到手上，纸张的质感也很棒，那种略带粗糙的纹理，翻起来沙沙作响，让人感觉这不是一本快餐式的读物，而是需要沉下心来细细品味的经典。内页的排版也十分考究，字体选用的是一种介于传统衬线体和现代无衬线体之间的字体，阅读起来既舒适又不失正式感。特别是那些复杂的数学公式，排版得非常清晰，没有出现任何因为压缩或错位导致的阅读障碍，这对于需要反复研读公式细节的读者来说，简直是福音。而且，书脊的装订非常牢固，即便经常翻阅到某个特定章节做标记，也不用担心书会散架。总而言之，从外在的感官体验来看，出版商在这本书的制作上绝对是下了大功夫的，体现了对内容本身应有的尊重。

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