Transformations of manifolds and applications to differential equations
著者
書誌事項
Transformations of manifolds and applications to differential equations
（Pitman monographs and surveys in pure and applied mathematics, 93）
Longman, 1998
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注記
Includes bibliographical references (p. [201]-206) and index
内容説明・目次
内容説明
The interaction between differential geometry and partial differential equations has been studied since the last century. This relationship is based on the fact that most of the local properties of manifolds are expressed in terms of partial differential equations. The correspondence between certain classes of manifolds and the associated differential equations can be useful in two ways. From our knowledge about the geometry of the manifolds we can obtain solutions to the equations. In particular it is important to study transformations of manifolds which preserve a geometric property, since the analytic interpretation of these transformations will provide mappings between the corresponding differential equations. Conversely, we can obtain geometric properties of the manifolds or even prove the non existence of certain geometric structures on manifolds from our knowledge of the differential equation. This kind of interaction between differential geometry and differential equations is the general theme of the book.
The author focuses on the role played by differential geometry in the study of differential equations, combining the geometric and analytic aspects of the theory, not only in the classical examples but also in results obtained since 1980, on integrable systems with an arbitrary number of independent variables. The book will be of interest to graduate students, researchers and mathematicians working in differential geometry, differential equations and mathematical physics.
目次
Preface
I Transformations of surfaces and applications
II Submanifolds of constant sectional curvature
III Pseudo-spherical geodesic congruences and applications
IV The generating equation
V The generating intrinsic equation
VI Laplace transformation in higher dimensions
References
Index
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