Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The present monograph grew out of the fifth set of Hermann Weyl Lectures, given by Professor Griffiths at the Institute for Advanced Study, Princeton, in fall 1974.In Chapter 1 the author discusses Emile Borel's proof and the classical Jensen theorem, order of growth of entire analytic sets, order functions for entire holomorphic mappings, classical indicators of orders of growth, and entire functions and varieties of finite order.Chapter 2 is devoted to the appearance of curvature, and Chapter 3 considers the defect relations. The author considers the lemma on the logarithmic derivative, R. Nevanlinna's proof of the defect relation, and refinements of the classical case.

Complex analysis --- Holomorphic mappings --- Applications holomorphes --- 517.53 --- Mappings, Holomorphic --- Functions of several complex variables --- Mappings (Mathematics) --- Functions of a complex variable --- Holomorphic mappings. --- 517.53 Functions of a complex variable --- Fonctions de plusieurs variables complexes --- Fonctions entières --- Functions, Entire --- Algebraic variety. --- Analytic function. --- Analytic set. --- Armand Borel. --- Big O notation. --- Canonical bundle. --- Cartesian coordinate system. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Chern class. --- Compact Riemann surface. --- Compact space. --- Complex analysis. --- Complex manifold. --- Complex projective space. --- Corollary. --- Counting. --- Curvature. --- Degeneracy (mathematics). --- Derivative. --- Differential form. --- Dimension. --- Divisor. --- Elementary proof. --- Entire function. --- Equation. --- Exponential growth. --- Gaussian curvature. --- Hermann Weyl. --- Hodge theory. --- Holomorphic function. --- Hyperplane. --- Hypersurface. --- Infinite product. --- Integral geometry. --- Invariant measure. --- Inverse problem. --- Jacobian matrix and determinant. --- Kähler manifold. --- Line bundle. --- Linear equation. --- Logarithmic derivative. --- Manifold. --- Meromorphic function. --- Modular form. --- Monograph. --- Nevanlinna theory. --- Nonlinear system. --- Phillip Griffiths. --- Picard theorem. --- Polynomial. --- Projective space. --- Q.E.D. --- Quantity. --- Ricci curvature. --- Riemann sphere. --- Scientific notation. --- Several complex variables. --- Special case. --- Stokes' theorem. --- Subset. --- Summation. --- Theorem. --- Theory. --- Uniformization theorem. --- Unit square. --- Volume form. --- Fonctions entières

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E. Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.

Differentiaalvergelijkingen [Hyperbolische ] --- Differential equations [Hyperbolic] --- Equations différentielles hyperboliques --- Symplectic manifolds --- Differential equations, Hyperbolic. --- Symplectic manifolds. --- Variétés symplectiques --- Equations différentielles hyperboliques --- Variétés symplectiques --- Manifolds, Symplectic --- Geometry, Differential --- Manifolds (Mathematics) --- Hyperbolic differential equations --- Differential equations, Partial --- Action (physics). --- Boundary value problem. --- Canonical form. --- Causal structure. --- Classical mechanics. --- Complex analysis. --- Configuration space. --- Conservative vector field. --- Conserved current. --- Conserved quantity. --- Continuum mechanics. --- Derivative. --- Diffeomorphism. --- Differentiable manifold. --- Differential geometry. --- Dimension. --- Dimensional analysis. --- Dirichlet's principle. --- Einstein field equations. --- Electromagnetic field. --- Equation. --- Equations of motion. --- Equivalence class. --- Error term. --- Euclidean space. --- Euler system. --- Euler's equations (rigid body dynamics). --- Euler–Lagrange equation. --- Existence theorem. --- Existential quantification. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Foliation. --- Fritz John. --- General relativity. --- Hamiltonian mechanics. --- Hamilton–Jacobi equation. --- Harmonic map. --- Hessian matrix. --- Holomorphic function. --- Hyperbolic partial differential equation. --- Hyperplane. --- Hypersurface. --- Identity element. --- Iteration. --- Iterative method. --- Lagrangian (field theory). --- Lagrangian. --- Legendre transformation. --- Lie algebra. --- Linear approximation. --- Linear differential equation. --- Linear map. --- Linear span. --- Linearity. --- Linearization. --- Maximum principle. --- Maxwell's equations. --- Nonlinear system. --- Open set. --- Ordinary differential equation. --- Orthogonal complement. --- Parameter. --- Partial differential equation. --- Phase space. --- Pointwise. --- Poisson bracket. --- Polynomial. --- Principal part. --- Principle of least action. --- Probability. --- Pullback bundle. --- Pullback. --- Quadratic form. --- Quantity. --- Requirement. --- Riemannian manifold. --- Second derivative. --- Simultaneous equations. --- Special case. --- State function. --- Stokes' theorem. --- Subset. --- Surjective function. --- Symplectic geometry. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Theoretical physics. --- Theory. --- Underdetermined system. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Zero of a function. --- Zero set.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The theory of D-modules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence, it is possible to attach geometric invariants, like the support and the characteristic variety, to representations of Lie groups. By considering D-modules on flag varieties, one obtains a simple classification of all irreducible admissible representations of reductive Lie groups. On the other hand, it is natural to study the representations realized by functions on pseudo-Riemannian symmetric spaces, i.e., spherical representations. The problem is then to describe the spherical representations among all irreducible ones, and to compute their multiplicities. This is the goal of this work, achieved fairly completely at least for the discrete series representations of reductive symmetric spaces. The book provides a general introduction to the theory of D-modules on flag varieties, and it describes spherical D-modules in terms of a cohomological formula. Using microlocalization of representations, the author derives a criterion for irreducibility. The relation between multiplicities and singularities is also discussed at length.Originally published in 1990.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Differentiable manifolds. --- D-modules. --- Representations of groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Modules (Algebra) --- Differential manifolds --- Manifolds (Mathematics) --- Affine space. --- Algebraic cycle. --- Algebraic element. --- Analytic function. --- Annihilator (ring theory). --- Automorphism. --- Banach space. --- Base change. --- Big O notation. --- Bijection. --- Bilinear form. --- Borel subgroup. --- Cartan subalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Commutator subgroup. --- Complexification (Lie group). --- Conjugacy class. --- Coproduct. --- Coset. --- Cotangent space. --- D-module. --- Derived category. --- Diagram (category theory). --- Differential operator. --- Dimension (vector space). --- Direct image functor. --- Discrete series representation. --- Disk (mathematics). --- Dot product. --- Double coset. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Euler operator. --- Existential quantification. --- Fibration. --- Function space. --- Functor. --- G-module. --- Gelfand pair. --- Generic point. --- Hilbert space. --- Holomorphic function. --- Homomorphism. --- Hyperfunction. --- Ideal (ring theory). --- Infinitesimal character. --- Inner automorphism. --- Invertible sheaf. --- Irreducibility (mathematics). --- Irreducible representation. --- Levi decomposition. --- Lie algebra. --- Line bundle. --- Linear algebraic group. --- Linear space (geometry). --- Manifold. --- Maximal compact subgroup. --- Maximal torus. --- Metric space. --- Module (mathematics). --- Moment map. --- Morphism. --- Noetherian ring. --- Open set. --- Presheaf (category theory). --- Principal series representation. --- Projective line. --- Projective object. --- Projective space. --- Projective variety. --- Reductive group. --- Riemannian geometry. --- Riemann–Hilbert correspondence. --- Right inverse. --- Ring (mathematics). --- Root system. --- Satake diagram. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Sphere. --- Square-integrable function. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subgroup. --- Summation. --- Surjective function. --- Symmetric space. --- Symplectic geometry. --- Tensor product. --- Theorem. --- Triangular matrix. --- Vector bundle. --- Volume form. --- Weyl group.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The subject matter of this work is an area of Lorentzian geometry which has not been heretofore much investigated: Do there exist Lorentzian manifolds all of whose light-like geodesics are periodic? A surprising fact is that such manifolds exist in abundance in (2 + 1)-dimensions (though in higher dimensions they are quite rare). This book is concerned with the deformation theory of M2,1 (which furnishes almost all the known examples of these objects). It also has a section describing conformal invariants of these objects, the most interesting being the determinant of a two dimensional "Floquet operator," invented by Paneitz and Segal.

Cosmology --- Geometry, Differential. --- Lorentz transformations. --- Mathematical models. --- Automorphism. --- Bijection. --- C0. --- Canonical form. --- Canonical transformation. --- Cauchy distribution. --- Causal structure. --- Cayley transform. --- Codimension. --- Cohomology. --- Cokernel. --- Compactification (mathematics). --- Complexification (Lie group). --- Computation. --- Conformal geometry. --- Conformal map. --- Conformal symmetry. --- Connected sum. --- Contact geometry. --- Corank. --- Covariant derivative. --- Covering space. --- Deformation theory. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Einstein field equations. --- Equation. --- Euler characteristic. --- Existential quantification. --- Fiber bundle. --- Fibration. --- Floquet theory. --- Four-dimensional space. --- Fourier integral operator. --- Fourier transform. --- Fundamental group. --- Geodesic. --- Hamilton–Jacobi equation. --- Hilbert space. --- Holomorphic function. --- Holomorphic vector bundle. --- Hyperfunction. --- Hypersurface. --- Integral curve. --- Integral geometry. --- Integral transform. --- Intersection (set theory). --- Invertible matrix. --- K-finite. --- Lagrangian (field theory). --- Lie algebra. --- Light cone. --- Linear map. --- Manifold. --- Maxima and minima. --- Minkowski space. --- Module (mathematics). --- Notation. --- One-parameter group. --- Parametrix. --- Parametrization. --- Principal bundle. --- Product metric. --- Pseudo-differential operator. --- Quadratic equation. --- Quadratic form. --- Quadric. --- Radon transform. --- Riemann surface. --- Riemannian manifold. --- Seifert fiber space. --- Sheaf (mathematics). --- Siegel domain. --- Simply connected space. --- Submanifold. --- Submersion (mathematics). --- Support (mathematics). --- Surjective function. --- Symplectic manifold. --- Symplectic vector space. --- Symplectomorphism. --- Tangent space. --- Tautology (logic). --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Two-dimensional space. --- Unit vector. --- Universal enveloping algebra. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Verma module. --- Volume form. --- X-ray transform.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Algebres de Clifford --- Clifford [Algebra's van ] --- Clifford algebras --- Fysica [Mathematische ] --- Fysica [Wiskundige ] --- Mathematische fysica --- Physics -- Mathematics --- Physics [Mathematical ] --- Physique -- Mathématiques --- Physique -- Méthodes mathématiques --- Wiskundige fysica --- Clifford, Algèbres de --- Spin, Nuclear --- Geometric algebras --- Clifford algebras. --- Spin geometry. --- Clifford, Algèbres de --- Spin geometry --- 514.76 --- Algebras, Linear --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Geometry of differentiable manifolds and of their submanifolds --- Global differential geometry --- Geometry --- Mathematical physics --- Topology --- Nuclear spin --- -Mathematics --- Géométrie --- Physique mathématique --- Spin nucléaire --- Topologie --- Mathematics --- Mathématiques --- Algebraic theory. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Binary icosahedral group. --- Binary octahedral group. --- Bundle metric. --- C*-algebra. --- Calabi conjecture. --- Calabi–Yau manifold. --- Cartesian product. --- Classification theorem. --- Clifford algebra. --- Cobordism. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Complete metric space. --- Complex manifold. --- Complex vector bundle. --- Complexification (Lie group). --- Covering space. --- Diffeomorphism. --- Differential topology. --- Dimension (vector space). --- Dimension. --- Dirac operator. --- Disk (mathematics). --- Dolbeault cohomology. --- Einstein field equations. --- Elliptic operator. --- Equivariant K-theory. --- Exterior algebra. --- Fiber bundle. --- Fixed-point theorem. --- Fourier inversion theorem. --- Fundamental group. --- Gauge theory. --- Geometry. --- Hilbert scheme. --- Holonomy. --- Homotopy sphere. --- Homotopy. --- Hyperbolic manifold. --- Induced homomorphism. --- Intersection form (4-manifold). --- Isomorphism class. --- J-invariant. --- K-theory. --- Kähler manifold. --- Laplace operator. --- Lie algebra. --- Lorentz covariance. --- Lorentz group. --- Manifold. --- Mathematical induction. --- Metric connection. --- Minkowski space. --- Module (mathematics). --- N-sphere. --- Operator (physics). --- Orthonormal basis. --- Principal bundle. --- Projective space. --- Pseudo-Riemannian manifold. --- Pseudo-differential operator. --- Quadratic form. --- Quaternion. --- Quaternionic projective space. --- Ricci curvature. --- Riemann curvature tensor. --- Riemannian geometry. --- Riemannian manifold. --- Ring homomorphism. --- Scalar curvature. --- Scalar multiplication. --- Sign (mathematics). --- Space form. --- Sphere theorem. --- Spin representation. --- Spin structure. --- Spinor bundle. --- Spinor field. --- Spinor. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Tangent bundle. --- Tangent space. --- Tensor calculus. --- Tensor product. --- Theorem. --- Topology. --- Unit disk. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Nuclear spin - - Mathematics --- -Clifford algebras.