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Mathematical No/ex, 27Originally published in 1981.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.

Riemannian manifolds. --- Minimal surfaces. --- Surfaces, Minimal --- Maxima and minima --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Differential geometry. Global analysis --- Addition. --- Analytic function. --- Branch point. --- Calculation. --- Cartesian coordinate system. --- Closed geodesic. --- Codimension. --- Coefficient. --- Compactness theorem. --- Compass-and-straightedge construction. --- Continuous function. --- Corollary. --- Counterexample. --- Covering space. --- Curvature. --- Curve. --- Decomposition theorem. --- Derivative. --- Differentiable manifold. --- Differential geometry. --- Disjoint union. --- Equation. --- Essential singularity. --- Estimation. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- First variation. --- Flat topology. --- Fundamental group. --- Geometric measure theory. --- Great circle. --- Homology (mathematics). --- Homotopy group. --- Homotopy. --- Hyperbolic function. --- Hypersurface. --- Integer. --- Line–line intersection. --- Manifold. --- Measure (mathematics). --- Minimal surface. --- Monograph. --- Natural number. --- Open set. --- Parameter. --- Partition of unity. --- Pointwise. --- Quantity. --- Regularity theorem. --- Riemann surface. --- Riemannian manifold. --- Scalar curvature. --- Scientific notation. --- Second fundamental form. --- Sectional curvature. --- Sequence. --- Sign (mathematics). --- Simply connected space. --- Smoothness. --- Sobolev inequality. --- Solid torus. --- Subgroup. --- Submanifold. --- Summation. --- Theorem. --- Topology. --- Two-dimensional space. --- Unit sphere. --- Upper and lower bounds. --- Varifold. --- Weak topology.

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The description for this book, Curvature and Betti Numbers. (AM-32), Volume 32, will be forthcoming.

Curvature. --- Geometry, Differential. --- Abelian integral. --- Affine connection. --- Algebraic operation. --- Almost periodic function. --- Analytic function. --- Arc length. --- Betti number. --- Coefficient. --- Compact space. --- Complex analysis. --- Complex conjugate. --- Complex dimension. --- Complex manifold. --- Conservative vector field. --- Constant curvature. --- Constant function. --- Continuous function. --- Convex set. --- Coordinate system. --- Covariance and contravariance of vectors. --- Covariant derivative. --- Derivative. --- Differential form. --- Differential geometry. --- Dimension (vector space). --- Dimension. --- Einstein manifold. --- Equation. --- Euclidean domain. --- Euclidean geometry. --- Euclidean space. --- Existential quantification. --- Geometry. --- Hausdorff space. --- Hypersphere. --- Killing vector field. --- Kähler manifold. --- Lie group. --- Manifold. --- Metric tensor (general relativity). --- Metric tensor. --- Mixed tensor. --- One-parameter group. --- Orientability. --- Partial derivative. --- Periodic function. --- Permutation. --- Quantity. --- Ricci curvature. --- Riemannian manifold. --- Scalar (physics). --- Sectional curvature. --- Self-adjoint. --- Special case. --- Subset. --- Summation. --- Symmetric tensor. --- Symmetrization. --- Tensor algebra. --- Tensor calculus. --- Tensor field. --- Tensor. --- Theorem. --- Torsion tensor. --- Two-dimensional space. --- Uniform convergence. --- Uniform space. --- Unit circle. --- Unit sphere. --- Unit vector. --- Vector field.

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The description for this book, Knot Groups. Annals of Mathematics Studies. (AM-56), Volume 56, will be forthcoming.

Topology --- 512 --- Algebra --- 512 Algebra --- Knot theory. --- Knots (Topology) --- Low-dimensional topology --- Abelian group. --- Alexander duality. --- Alexander polynomial. --- Algebraic theory. --- Algorithm. --- Analytic continuation. --- Associative property. --- Automorphism. --- Axiom. --- Bijection. --- Binary relation. --- Calculation. --- Central series. --- Characterization (mathematics). --- Cobordism. --- Coefficient. --- Cohomology. --- Combinatorics. --- Commutator subgroup. --- Complete theory. --- Computation. --- Conjugacy class. --- Conjugate element (field theory). --- Connected space. --- Connectedness. --- Coprime integers. --- Coset. --- Covering space. --- Curve. --- Cyclic group. --- Dehn's lemma. --- Determinant. --- Diagonalization. --- Diagram (category theory). --- Dimension. --- Direct product. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Euler characteristic. --- Existential quantification. --- Fiber bundle. --- Finite group. --- Finitely generated module. --- Frattini subgroup. --- Free abelian group. --- Fundamental group. --- Geometry. --- Group ring. --- Group theory. --- Group with operators. --- Hausdorff space. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Homotopy group. --- Homotopy. --- Identity matrix. --- Inner automorphism. --- Interior (topology). --- Intersection number (graph theory). --- Knot group. --- Linear combination. --- Manifold. --- Mathematical induction. --- Monomorphism. --- Morphism. --- Morse theory. --- Natural transformation. --- Non-abelian group. --- Normal subgroup. --- Orientability. --- Permutation. --- Polynomial. --- Presentation of a group. --- Principal ideal domain. --- Principal ideal. --- Root of unity. --- Semigroup. --- Simplicial complex. --- Simply connected space. --- Special case. --- Square matrix. --- Subgroup. --- Subset. --- Summation. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Topology. --- Torus knot. --- Transfinite number. --- Trefoil knot. --- Trichotomy (mathematics). --- Trivial group. --- Triviality (mathematics). --- Two-dimensional space. --- Unit vector. --- Wreath product.

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Locally symmetric spaces are generalizations of spaces of constant curvature. In this book the author presents the proof of a remarkable phenomenon, which he calls "strong rigidity": this is a stronger form of the deformation rigidity that has been investigated by Selberg, Calabi-Vesentini, Weil, Borel, and Raghunathan.The proof combines the theory of semi-simple Lie groups, discrete subgroups, the geometry of E. Cartan's symmetric Riemannian spaces, elements of ergodic theory, and the fundamental theorem of projective geometry as applied to Tit's geometries. In his proof the author introduces two new notions having independent interest: one is "pseudo-isometries"; the other is a notion of a quasi-conformal mapping over the division algebra K (K equals real, complex, quaternion, or Cayley numbers). The author attempts to make the account accessible to readers with diverse backgrounds, and the book contains capsule descriptions of the various theories that enter the proof.

Differential geometry. Global analysis --- Riemannian manifolds --- Symmetric spaces --- Rigidity (Geometry) --- 512 --- Lie groups --- Geometric rigidity --- Rigidity theorem --- Discrete geometry --- Spaces, Symmetric --- Geometry, Differential --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Manifolds (Mathematics) --- Groups, Lie --- Lie algebras --- Topological groups --- Algebra --- Lie groups. --- Riemannian manifolds. --- Symmetric spaces. --- Rigidity (Geometry). --- 512 Algebra --- Addition. --- Adjoint representation. --- Affine space. --- Approximation. --- Automorphism. --- Axiom. --- Big O notation. --- Boundary value problem. --- Cohomology. --- Compact Riemann surface. --- Compact space. --- Conjecture. --- Constant curvature. --- Corollary. --- Counterexample. --- Covering group. --- Covering space. --- Curvature. --- Diameter. --- Diffeomorphism. --- Differentiable function. --- Dimension. --- Direct product. --- Division algebra. --- Ergodicity. --- Erlangen program. --- Existence theorem. --- Exponential function. --- Finitely generated group. --- Fundamental domain. --- Fundamental group. --- Geometry. --- Half-space (geometry). --- Hausdorff distance. --- Hermitian matrix. --- Homeomorphism. --- Homomorphism. --- Hyperplane. --- Identity matrix. --- Inner automorphism. --- Isometry group. --- Jordan algebra. --- Matrix multiplication. --- Metric space. --- Morphism. --- Möbius transformation. --- Normal subgroup. --- Normalizing constant. --- Partially ordered set. --- Permutation. --- Projective space. --- Riemann surface. --- Riemannian geometry. --- Sectional curvature. --- Self-adjoint. --- Set function. --- Smoothness. --- Stereographic projection. --- Subgroup. --- Subset. --- Summation. --- Symmetric space. --- Tangent space. --- Tangent vector. --- Theorem. --- Topology. --- Tubular neighborhood. --- Two-dimensional space. --- Unit sphere. --- Vector group. --- Weyl group. --- Riemann, Variétés de --- Lie, Groupes de --- Geometrie differentielle globale --- Varietes riemanniennes

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Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

Differential geometry. Global analysis --- Bifurcation theory. --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Addition. --- Algebraic equation. --- Analytic function. --- Analytic manifold. --- Atmospheric pressure. --- Banach space. --- Bernhard Riemann. --- Bifurcation diagram. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Boundedness. --- Canonical form. --- Cartesian coordinate system. --- Codimension. --- Compact operator. --- Complex analysis. --- Complex conjugate. --- Complex number. --- Connected space. --- Coordinate system. --- Corollary. --- Curvature. --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Embedding. --- Equation. --- Euclidean division. --- Euler equations (fluid dynamics). --- Existential quantification. --- First principle. --- Fredholm operator. --- Froude number. --- Functional analysis. --- Hilbert space. --- Homeomorphism. --- Implicit function theorem. --- Integer. --- Linear algebra. --- Linear function. --- Linear independence. --- Linear map. --- Linear programming. --- Linear space (geometry). --- Linear subspace. --- Linearity. --- Linearization. --- Metric space. --- Morse theory. --- Multilinear form. --- N0. --- Natural number. --- Neumann series. --- Nonlinear functional analysis. --- Nonlinear system. --- Numerical analysis. --- Open mapping theorem (complex analysis). --- Operator (physics). --- Ordinary differential equation. --- Parameter. --- Parametrization. --- Partial differential equation. --- Permutation group. --- Permutation. --- Polynomial. --- Power series. --- Prime number. --- Proportionality (mathematics). --- Pseudo-differential operator. --- Puiseux series. --- Quantity. --- Real number. --- Resultant. --- Singularity theory. --- Skew-symmetric matrix. --- Smoothness. --- Solution set. --- Special case. --- Standard basis. --- Sturm–Liouville theory. --- Subset. --- Symmetric bilinear form. --- Symmetric group. --- Taylor series. --- Taylor's theorem. --- Theorem. --- Total derivative. --- Two-dimensional space. --- Union (set theory). --- Variable (mathematics). --- Vector space. --- Zero of a function.