The Laplacian on a Riemannian Manifold 2024 pdf epub mobi 电子书

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发表于2024-12-18

The Laplacian on a Riemannian Manifold 2024 pdf epub mobi 电子书

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出版者:Cambridge University Press

作者:Steven Rosenberg

出品人:

页数:188

译者:

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

The Laplacian on a Riemannian Manifold 2024 pdf epub mobi 电子书图书描述

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.

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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

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