Softcover ISBN:  9780821807071 
Product Code:  CBMS/57 
List Price:  $27.00 
Individual Price:  $21.60 
Electronic ISBN:  9781470424190 
Product Code:  CBMS/57.E 
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Book DetailsCBMS Regional Conference Series in MathematicsVolume: 57; 1985; 55 ppMSC: Primary 53; Secondary 58;
These notes were the basis for a series of ten lectures given in January 1984 at Polytechnic Institute of New York under the sponsorship of the Conference Board of the Mathematical Sciences and the National Science Foundation. The lectures were aimed at mathematicians who knew either some differential geometry or partial differential equations, although others could understand the lectures.
Author's Summary:Given a Riemannian Manifold \((M,g)\) one can compute the sectional, Ricci, and scalar curvatures. In other special circumstances one also has mean curvatures, holomorphic curvatures, etc. The inverse problem is, given a candidate for some curvature, to determine if there is some metric \(g\) with that as its curvature. One may also restrict ones attention to a special class of metrics, such as Kähler or conformal metrics, or those coming from an embedding. These problems lead one to (try to) solve nonlinear partial differential equations. However, there may be topological or analytic obstructions to solving these equations. A discussion of these problems thus requires a balanced understanding between various existence and nonexistence results.
The intent of this volume is to give an uptodate survey of these questions, including enough background, so that the current research literature is accessible to mathematicians who are not necessarily experts in PDE or differential geometry.
The intended audience is mathematicians and graduate students who know either PDE or differential geometry at roughly the level of an intermediate graduate course.Readership 
Table of Contents

Chapters

Gaussian Curvature

Scalar Curvature

Ricci Curvature

Boundary Value Problems

Some Open Problems


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These notes were the basis for a series of ten lectures given in January 1984 at Polytechnic Institute of New York under the sponsorship of the Conference Board of the Mathematical Sciences and the National Science Foundation. The lectures were aimed at mathematicians who knew either some differential geometry or partial differential equations, although others could understand the lectures.
Author's Summary:Given a Riemannian Manifold \((M,g)\) one can compute the sectional, Ricci, and scalar curvatures. In other special circumstances one also has mean curvatures, holomorphic curvatures, etc. The inverse problem is, given a candidate for some curvature, to determine if there is some metric \(g\) with that as its curvature. One may also restrict ones attention to a special class of metrics, such as Kähler or conformal metrics, or those coming from an embedding. These problems lead one to (try to) solve nonlinear partial differential equations. However, there may be topological or analytic obstructions to solving these equations. A discussion of these problems thus requires a balanced understanding between various existence and nonexistence results.
The intent of this volume is to give an uptodate survey of these questions, including enough background, so that the current research literature is accessible to mathematicians who are not necessarily experts in PDE or differential geometry.
The intended audience is mathematicians and graduate students who know either PDE or differential geometry at roughly the level of an intermediate graduate course.

Chapters

Gaussian Curvature

Scalar Curvature

Ricci Curvature

Boundary Value Problems

Some Open Problems