The Laplacian on a Riemannian Manifold

出版者:Cambridge University Press

作者:Steven Rosenberg

页数:188

出版时间:1997-1-28

价格:USD 62.00

装帧:Paperback

isbn号码:9780521468312

丛书系列:London Mathematical Society Student Texts

图书标签:黎曼几何数学几何分析Mathematics微分几何7几何Riemann_Geometry机器学习

This text on analysis of Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The Atiyah-Singer index theorem and its applications are developed (without complete proofs) via the heat equation method. Zeta functions for Laplacians and analytic torsion are also treated, and the recently uncovered relation between index theory and analytic torsion is laid out. The text is aimed at students who have had a first course in differentiable manifolds, and the Riemannian geometry used is developed from the beginning. There are over 100 exercises with hints.

用户评价

Atiyah-Singer指标定理。微分拓扑解释光滑流形的整体性质，微分几何研究整体（测地线）和局部（曲率）的关系。黎曼流形上的距离诱导的拓扑和原流形拓扑等价，则测地线的度量就是流形上每一点测地凸域拓扑。Sturm-Liouville 推广了傅里叶级数。极小曲线不一定存在，则定义曲线最小上界。拉普拉斯算子决定黎曼度量，反之也对。霍奇星算子和斯托克斯定理导致外微分的伴随算子与坐标无关，紧致集上所有光滑函数且带内积的空间完备化是希尔伯特空间。热流的长时间是拓扑相关，而短时间是恒等算子。热核逼近狄拉克函数被局部的黎曼流形

Atiyah-Singer指标定理。微分拓扑解释光滑流形的整体性质，微分几何研究整体（测地线）和局部（曲率）的关系。黎曼流形上的距离诱导的拓扑和原流形拓扑等价，则测地线的度量就是流形上每一点测地凸域拓扑。Sturm-Liouville 推广了傅里叶级数。极小曲线不一定存在，则定义曲线最小上界。拉普拉斯算子决定黎曼度量，反之也对。霍奇星算子和斯托克斯定理导致外微分的伴随算子与坐标无关，紧致集上所有光滑函数且带内积的空间完备化是希尔伯特空间。热流的长时间是拓扑相关，而短时间是恒等算子。热核逼近狄拉克函数被局部的黎曼流形

体量很小，却是很完整的一本书。指标定理的漂亮与重要性都展现出来了，很多内容的概观和直观也不错。以前Peter Petersen的黎曼几何都读不下去，大概是我真的很讨厌读事无巨细的教材吧…

体量很小，却是很完整的一本书。指标定理的漂亮与重要性都展现出来了，很多内容的概观和直观也不错。以前Peter Petersen的黎曼几何都读不下去，大概是我真的很讨厌读事无巨细的教材吧…

Atiyah-Singer指标定理。微分拓扑解释光滑流形的整体性质，微分几何研究整体（测地线）和局部（曲率）的关系。黎曼流形上的距离诱导的拓扑和原流形拓扑等价，则测地线的度量就是流形上每一点测地凸域拓扑。Sturm-Liouville 推广了傅里叶级数。极小曲线不一定存在，则定义曲线最小上界。拉普拉斯算子决定黎曼度量，反之也对。霍奇星算子和斯托克斯定理导致外微分的伴随算子与坐标无关，紧致集上所有光滑函数且带内积的空间完备化是希尔伯特空间。热流的长时间是拓扑相关，而短时间是恒等算子。热核逼近狄拉克函数被局部的黎曼流形