Aspects Of Overdetermined Systems Of Partial Differential Equations In Projective And Conformal Geometry

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Aspects Of Overdetermined Systems Of Partial Differential Equations In Projective And Conformal Geometry

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Author : Matthew James Randall
language : en
Publisher:
Release Date : 2013

Aspects Of Overdetermined Systems Of Partial Differential Equations In Projective And Conformal Geometry written by Matthew James Randall and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Conformal geometry categories.

This thesis discusses aspects of overdetermined systems of partial differential equations (PDEs) in projective and conformal geometry. The first part deals with projective differential geometry. A projective surface is a 2-dimensional smooth manifold equipped with a projective structure i.e. a class of torsion-free affine connections that have the same geodesics as unparameterised curves. Given any projective surface we can ask whether it admits a torsion-free affine connection (in its projective class) that has skew-symmetric Ricci tensor. This is equivalent to solving a particular overdetermined system of semi-linear partial differential equations. It turns out that there are local obstructions to solving the system of PDEs in two dimensions. These obstructions are constructed out of local invariants of the projective structure. We give examples of projective surfaces that admit skew-symmetric Ricci tensor and examples that do not because of nonvanishing obstructions. We relate projective surfaces admitting skew-symmetric Ricci tensor to 3-webs in 2 dimensions. We also give examples of projective structures in higher dimensions that admit skew-symmetric Ricci tensor. The second part of the thesis deals with conformal differential geometry. On Mobius surfaces introduced in [5], we can define an analogous overdetermined system of semi-linear PDEs as in the projective case. This is called the scalar-flat Mobius Einstein-Weyl equation and is conformally invariant. We derive local algebraic constraints for Mobius surfaces to admit a solution to this equation and give local obstructions. These obstructions are similarly constructed out of local conformal invariants of the Mobius structure. Again we provide examples of Mobius surfaces that admit a solution and examples that do not because of non-vanishing obstructions. Finally, we also investigate the conformally Einstein equation on Mobius surfaces and derive obstructions. In contrast to the previous two equations, the conformally Einstein equation is linear.

Symmetries And Overdetermined Systems Of Partial Differential Equations

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Author : Michael Eastwood
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-04-23

Symmetries And Overdetermined Systems Of Partial Differential Equations written by Michael Eastwood and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-04-23 with Mathematics categories.

This three-week summer program considered the symmetries preserving various natural geometric structures. There are two parts to the proceedings. The articles in the first part are expository but all contain significant new material. The articles in the second part are concerned with original research. All articles were thoroughly refereed and the range of interrelated work ensures that this will be an extremely useful collection.

Nonlinear Partial Differential Equations In Differential Geometry

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Author : Robert Hardt
language : en
Publisher: American Mathematical Soc.
Release Date : 1996

Nonlinear Partial Differential Equations In Differential Geometry written by Robert Hardt and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Mathematics categories.

This book contains lecture notes of minicourses at the Regional Geometry Institute at Park City, Utah, in July 1992. Presented here are surveys of breaking developments in a number of areas of nonlinear partial differential equations in differential geometry. The authors of the articles are not only excellent expositors, but are also leaders in this field of research. All of the articles provide in-depth treatment of the topics and require few prerequisites and less background than current research articles.

Exterior Differential Systems And Euler Lagrange Partial Differential Equations

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Author : Robert Bryant
language : en
Publisher: University of Chicago Press
Release Date : 2003-07

Exterior Differential Systems And Euler Lagrange Partial Differential Equations written by Robert Bryant and has been published by University of Chicago Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-07 with Mathematics categories.

In Exterior Differential Systems, the authors present the results of their ongoing development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincaré-Cartan forms. They also cover certain aspects of the theory of exterior differential systems, which provides the language and techniques for the entire study. Because it plays a central role in uncovering geometric properties of differential equations, the method of equivalence is particularly emphasized. In addition, the authors discuss conformally invariant systems at length, including results on the classification and application of symmetries and conservation laws. The book also covers the Second Variation, Euler-Lagrange PDE systems, and higher-order conservation laws. This timely synthesis of partial differential equations and differential geometry will be of fundamental importance to both students and experienced researchers working in geometric analysis.

Involutive Hyperbolic Differential Systems

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Author : Deane Yang
language : en
Publisher: American Mathematical Soc.
Release Date : 1987

Involutive Hyperbolic Differential Systems written by Deane Yang and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Mathematics categories.

The Cartan-Kähler theorem demonstrates that certain real analytic exterior differential systems--ones which Cartan dubbed "involutive"--admit local real analytic solutions. The proof reduces the system to a set of determined systems of PDE's which are then solved using the Cauchy-Kovalevski theorem. The subtlety in the theorem and in the definition of involutivity lies in checking that the "solution" obtained from the determined systems does in fact satisfy the original differential system.

An Introduction To Overdetermined Systems Of Partial Differential Equations

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Author : Hubert Goldschmidt
language : en
Publisher:
Release Date : 1986

An Introduction To Overdetermined Systems Of Partial Differential Equations written by Hubert Goldschmidt and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1986 with categories.

Overdetermined Systems Of Linear Partial Differential Equations

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Author : Donald Clayton Spencer
language : en
Publisher:
Release Date : 1968

Overdetermined Systems Of Linear Partial Differential Equations written by Donald Clayton Spencer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1968 with Differential equations, Partial categories.

Overdetermined Systems Of Analytic Partial Differential Equations

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Author : Hubert Leopold Goldschmidt
language : en
Publisher:
Release Date : 1967

Overdetermined Systems Of Analytic Partial Differential Equations written by Hubert Leopold Goldschmidt and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1967 with categories.

Formal Properties Of Over Determined Systems Of Linear Partial Differential Equations

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Author : Daniel Gray Quillen
language : en
Publisher:
Release Date : 1964

Formal Properties Of Over Determined Systems Of Linear Partial Differential Equations written by Daniel Gray Quillen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1964 with categories.

Differential Systems And Isometric Embeddings

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Author : Phillip Griffiths
language : en
Publisher:
Release Date : 1987

Differential Systems And Isometric Embeddings written by Phillip Griffiths and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Science categories.

The theory of exterior differential systems provides a framework for systematically addressing the typically non-linear, and frequently overdetermined, partial differential equations that arise in differential geometry. Adaptation of the techniques of microlocalization to differential systems have led to recent activity on the foundations of the theory; in particular, the fundamental role of the characteristic variety in geometric problems is now clearly established. In this book the general theory is explained in a relatively quick and concrete manner, and then this general theory is applied to the recent developments in the classical problem of isometric embeddings of Riemannian manifolds.