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In this monograph the authors redevelop the theory systematically using two different approaches. A general mechanism for the deformation of structures on manifolds was developed by Donald Spencer ten years ago. A new version of that theory, based on the differential calculus in the analytic spaces of Grothendieck, was recently given by B. Malgrange. The first approach adopts Malgrange's idea in defining jet sheaves and linear operators, although the brackets and the non-linear theory arc treated in an essentially different manner. The second approach is based on the theory of derivations, and its relationship to the first is clearly explained. The introduction describes examples of Lie equations and known integrability theorems, and gives applications of the theory to be developed in the following chapters and in the subsequent volume.
Differential geometry. Global analysis --- Lie groups --- Lie algebras --- Differential equations --- Groupes de Lie --- Algèbres de Lie --- Equations différentielles --- 514.76 --- Groups, Lie --- Symmetric spaces --- Topological groups --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Equations, Differential --- Bessel functions --- Calculus --- Geometry of differentiable manifolds and of their submanifolds --- Differential equations. --- Lie algebras. --- Lie groups. --- 517.91 Differential equations --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Algèbres de Lie --- Equations différentielles --- 517.91. --- Numerical solutions --- Surfaces, Deformation of --- Surfaces (mathématiques) --- Déformation --- Pseudogroups. --- Pseudogroupes (mathématiques) --- 517.91 --- Adjoint representation. --- Adjoint. --- Affine transformation. --- Alexander Grothendieck. --- Analytic function. --- Associative algebra. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Bundle map. --- Category of topological spaces. --- Cauchy–Riemann equations. --- Coefficient. --- Commutative diagram. --- Commutator. --- Complex conjugate. --- Complex group. --- Complex manifold. --- Computation. --- Conformal map. --- Continuous function. --- Coordinate system. --- Corollary. --- Cotangent bundle. --- Curvature tensor. --- Deformation theory. --- Derivative. --- Diagonal. --- Diffeomorphism. --- Differentiable function. --- Differential form. --- Differential operator. --- Differential structure. --- Direct proof. --- Direct sum. --- Ellipse. --- Endomorphism. --- Equation. --- Exact sequence. --- Exactness. --- Existential quantification. --- Exponential function. --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Fibration. --- Frame bundle. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Group isomorphism. --- Groupoid. --- Holomorphic function. --- Homeomorphism. --- Integer. --- J-invariant. --- Jacobian matrix and determinant. --- Jet bundle. --- Linear combination. --- Linear map. --- Manifold. --- Maximal ideal. --- Model category. --- Morphism. --- Nonlinear system. --- Open set. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Pointwise. --- Presheaf (category theory). --- Pseudo-differential operator. --- Pseudogroup. --- Quantity. --- Regular map (graph theory). --- Requirement. --- Riemann surface. --- Right inverse. --- Scalar multiplication. --- Sheaf (mathematics). --- Special case. --- Structure tensor. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor product. --- Theorem. --- Torsion tensor. --- Transpose. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume element. --- Surfaces (mathématiques) --- Déformation --- Analyse sur une variété
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Partial differential equations --- 517.9 --- Differential equations, Partial --- Pseudogroup structures, Deformation of --- Deformation of pseudogroup structures --- Pseudogroups --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Differential equations, Partial. --- Pseudogroup structures, Deformation of. --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Surfaces (mathématiques) --- Surfaces, Deformation of --- Déformation --- Pseudogroups. --- Pseudogroupes (mathématiques) --- Déformation. --- Surfaces (mathématiques) --- Déformation --- Equations aux derivees partielles lineaires
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The description for this book, Contributions to the Theory of Riemann Surfaces. (AM-30), Volume 30, will be forthcoming.
Riemann surfaces. --- Abelian integral. --- Algebraic curve. --- Algebraic function. --- Algebraic geometry. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Asymptotic formula. --- Automorphic function. --- Automorphism. --- Banach algebra. --- Bernhard Riemann. --- Boundary value problem. --- Bounded set (topological vector space). --- Coefficient. --- Compact Riemann surface. --- Compactification (mathematics). --- Complete metric space. --- Complex analysis. --- Complex manifold. --- Conformal map. --- Degeneracy (mathematics). --- Differential equation. --- Differential geometry. --- Differential of the first kind. --- Dimension (vector space). --- Dirichlet integral. --- Dirichlet problem. --- Dirichlet's principle. --- Divisor (algebraic geometry). --- Eigenvalues and eigenvectors. --- Elliptic function. --- Elliptic partial differential equation. --- Equation. --- Existence theorem. --- Existential quantification. --- Explicit formulae (L-function). --- Extremal length. --- Function (mathematics). --- Functional equation. --- Fundamental group. --- Fundamental theorem. --- Geometric function theory. --- Green's function. --- Harmonic conjugate. --- Harmonic function. --- Harmonic measure. --- Holomorphic function. --- Hyperbolic geometry. --- Hypergeometric function. --- Integral equation. --- Intersection (set theory). --- Interval (mathematics). --- Isometry. --- Isoperimetric inequality. --- Jordan curve theorem. --- Kähler manifold. --- Laplace's equation. --- Lebesgue integration. --- Linear differential equation. --- Linear map. --- Linear space (geometry). --- Mathematical physics. --- Mathematical theory. --- Mathematics. --- Meromorphic function. --- Metric space. --- Minkowski space. --- Operator (physics). --- Ordinary differential equation. --- Parametric equation. --- Parity (mathematics). --- Partial differential equation. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Quadratic differential. --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Riemann–Roch theorem. --- Ring (mathematics). --- Scalar (physics). --- Sign (mathematics). --- Simultaneous equations. --- Special case. --- Surjective function. --- Tensor density. --- Theorem. --- Theory of equations. --- Theory. --- Topology. --- Uniformization theorem. --- Uniformization. --- Uniqueness theorem. --- Variable (mathematics). --- Weierstrass theorem.
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