Differential Geometry and Lie Groups: A Computational Perspective

Gallier, Jean, Quaintance, Jocelyn

Differential Geometry and Lie Groups: A Computational Perspective

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商品描述

This textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.

Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.

Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.

Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors' companion volume Differential Geometry and Lie Groups: A Second Course.

商品描述(中文翻譯)

這本教科書提供了一個微分幾何的介紹，針對對現代幾何處理感興趣的讀者。作者從基礎的大學先修課程開始，從頭開始發展流形理論和李群；隨後介紹了黎曼幾何的基本主題，最終涵蓋了支撐流形優化技術的理論。在計算機視覺、機器人和機器學習領域工作的學生和專業人士將會喜歡這本書，因為它提供了進入許多現代應用背後的數學概念的途徑。

從矩陣指數開始，本書介紹了李群和群作用。接著介紹了流形、切空間和余切空間；一章關於通過黏合數據構造流形的內容對於從3D網格重建表面特別相關。向量場和基本點集拓撲為本書的第二部分鋪墊，該部分重點介紹了黎曼幾何。

關於黎曼流形的章節包括黎曼度量、测地线和曲率。隨後的主題包括浸入、李群上的曲率和對數歐幾里得框架。最後一章突出了自然紅副同構流形和對稱空間，揭示了將重要的優化技術推廣到黎曼流形所需的機制。書中包含了練習題，以及深入探討更理論性主題的可選章節。

《微分幾何和李群：計算透視》以獨特的角度介紹了微分幾何，適合那些對現代計算應用背後的理論感興趣的人。無論是在課堂上使用還是獨立學習，這本書都適合學生和專業人士；只需要具備微積分和線性代數的背景知識。希望繼續學習更高級主題的讀者將會喜歡作者的同伴著作《微分幾何和李群：第二門課》。

作者簡介

Jean Gallier is Professor of Computer and Information Science at the University of Pennsylvania, Philadelphia. His research interests include geometry and its applications, geometric modeling, and differential geometry. He is also a member of the University of Pennsylvania's Department of Mathematics, and its Center for Human Modelling and Simulation.

Jocelyn Quaintance is postdoctoral researcher at the University of Pennsylvania who has contributed to the fields of combinatorial identities and power product expansions. Her recent mathematical books investigate the interplay between mathematics and computer science. Covering areas as diverse as differential geometry, linear algebra, optimization theory, and Fourier analysis, her writing illuminates the mathematics behind topics relevant to engineering, computer vision, and robotics.

作者簡介(中文翻譯)

Jean Gallier是賓夕法尼亞大學費城校區的計算機與資訊科學教授。他的研究興趣包括幾何學及其應用、幾何建模和微分幾何。他還是賓夕法尼亞大學數學系和人體建模與模擬中心的成員。

Jocelyn Quaintance是賓夕法尼亞大學的博士後研究員，她對組合恆等式和冪次展開等領域做出了貢獻。她最近的數學書籍探討了數學與計算機科學之間的相互作用。她的著作涵蓋了微分幾何、線性代數、最佳化理論和傅立葉分析等多個領域，闡明了與工程、計算機視覺和機器人相關的主題背後的數學原理。

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