## Partial Differential RelationsThe classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature. Law abiding functions, which satisfy such an equation, are very rare in the space of all admissible functions (regardless of a particular topology in a function space). Moreover, some additional (like initial or boundary) conditions often insure the uniqueness of solutions. The existence of these is usually established with some apriori estimates which locate a possible solution in a given function space. We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather than in physics. Our equations are, for the most part, undetermined (or, at least, behave like those) and their solutions are rather dense in spaces of functions. We solve and classify solutions of these equations by means of direct (and not so direct) geometric constructions. Our exposition is elementary and the proofs of the basic results are selfcontained. However, there is a number of examples and exercises (of variable difficulty), where the treatment of a particular equation requires a certain knowledge of pertinent facts in the surrounding field. The techniques we employ, though quite general, do not cover all geometrically interesting equations. The border of the unexplored territory is marked by a number of open questions throughout the book. |

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### Contents

Section 1 | 1 |

Section 2 | 25 |

Section 3 | 28 |

Section 4 | 48 |

Section 5 | 64 |

Section 6 | 101 |

Section 7 | 115 |

Section 8 | 128 |

Section 17 | 198 |

Section 18 | 202 |

Section 19 | 207 |

Section 20 | 221 |

Section 21 | 224 |

Section 22 | 227 |

Section 23 | 233 |

Section 24 | 235 |

Section 9 | 135 |

Section 10 | 151 |

Section 11 | 160 |

Section 12 | 168 |

Section 13 | 171 |

Section 14 | 174 |

Section 15 | 177 |

Section 16 | 189 |

Section 25 | 258 |

Section 26 | 277 |

Section 27 | 280 |

Section 28 | 284 |

Section 29 | 288 |

Section 30 | 350 |

Section 31 | 359 |

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### Common terms and phrases

admits an isometric applies arbitrary assume C°-close called closed manifold codim codimension compact subsets condition conformai connected constant continuous map Conv Corollary curvature Cx-immersion defined deformation denote derivatives diffeomorphisms diffeotopy differential operator differential relation dimension dimensional embedding equations Euclidean example Exercise exists extension fiber fibration finite flexible form g free isometric free maps Furthermore given Hence holonomic section homomorphism hyperbolic hyperplane hypersurface i-principle implicit function theorem implies inequality infinitesimally inversion isometric C2-immersion isometric immersion isometric map Lemma linear map F Math metric g microflexible n-dimensional normal obviously open manifold open subset parallelizable parallelizable manifold parametric polynomial projection Proof Prove quadratic real analytic Riemannian manifold Riemannian metric satisfy sheaf short map Show singular small neighborhood smooth solutions space strictly short subbundle submanifold subspace symplectic symplectic manifold tangent topological transversal vector bundle vector fields zero

### References to this book

Chiral Algebras, Volume 51 Alexander Beilinson,Vladimir Drinfeld,V. G. Drinfeld No preview available - 2004 |