Differential Geometry and Lie Groups: A Computational Perspective
暫譯: 微分幾何與李群：計算視角

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針對對現代幾何處理感興趣的讀者的微分幾何入門。從基本的本科先修知識開始，作者從零開始發展流形理論和李群；接著介紹黎曼幾何的基本主題，最終達到支撐流形優化技術的理論。從事計算機視覺、機器人技術和機器學習的學生和專業人士將會欣賞這條通往許多現代應用背後數學概念的道路。

本書從矩陣指數開始，首先介紹李群和群作用。接下來是流形、切空間和共切空間；有關從粘合數據構造流形的章節對於從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路，該部分專注於黎曼幾何。

有關黎曼流形的章節涵蓋了黎曼度量、測地線和曲率。隨後的主題包括潛射、李群上的曲率以及對數歐幾里得框架。最後一章強調自然還原的均勻流形和對稱空間，揭示了將重要優化技術推廣到黎曼流形所需的機制。全書包含練習題，並有可選的部分深入探討更理論的主題。

《微分幾何與李群：計算視角》為那些對現代計算應用背後的理論感興趣的人提供了一個獨特且易於理解的微分幾何視角。本書同樣適合用於課堂教學或獨立學習，將吸引學生和專業人士；僅假設具備微積分和線性代數的背景。希望繼續深入更高級主題的讀者將會欣賞作者的伴隨著作《微分幾何與李群：第二課程》。

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（University of Pennsylvania）計算機與信息科學的教授。他的研究興趣包括幾何及其應用、幾何建模和微分幾何。他也是賓夕法尼亞大學數學系的成員，以及人類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，對組合身份和冪積展開領域做出了貢獻。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。涵蓋微分幾何、線性代數、優化理論和傅里葉分析等多個領域，她的著作闡明了與工程、計算機視覺和機器人技術相關主題背後的數學。

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