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Relativity --- Differential calculus --- Gravitational fields --- Covariant derivative

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Wave mechanics --- Elastic waves in solids --- Analytical mechanics --- Solid state physics --- Covariant derivative in geometry

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The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.

Algebraic geometry --- Algebraïsche meetkunde --- Geometry [Algebraic ] --- Géométrie algébrique --- Meetkunde [Algebraïsche ] --- Riemann-Roch theorema's --- Riemann-Roch thoerems --- Theoremes de Riemann-Roch --- Geometry, Algebraic. --- Riemann-Roch theorems. --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry, Algebraic --- Geometry --- Addition. --- Adjoint. --- Alexander Grothendieck. --- Algebraic geometry. --- Analytic torsion. --- Arakelov theory. --- Asymptote. --- Asymptotic expansion. --- Asymptotic formula. --- Big O notation. --- Cartesian coordinate system. --- Characteristic class. --- Chern class. --- Chow group. --- Closed immersion. --- Codimension. --- Coherent sheaf. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Covariant derivative. --- Curvature. --- Derivative. --- Determinant. --- Diagonal. --- Differentiable manifold. --- Differential form. --- Dimension (vector space). --- Divisor. --- Domain of a function. --- Dual basis. --- E6 (mathematics). --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Exact sequence. --- Exponential function. --- Generic point. --- Heat kernel. --- Injective function. --- Intersection theory. --- K-group. --- Levi-Civita connection. --- Line bundle. --- Linear algebra. --- Local coordinates. --- Mathematical induction. --- Morphism. --- Natural number. --- Neighbourhood (mathematics). --- Parameter. --- Projective space. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemannian manifold. --- Riemann–Roch theorem. --- Self-adjoint operator. --- Smoothness. --- Sobolev space. --- Stochastic calculus. --- Summation. --- Supertrace. --- Theorem. --- Transition function. --- Upper half-plane. --- Vector bundle. --- Volume form.

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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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This work is a fresh presentation of the Ahlfors-Weyl theory of holomorphic curves that takes into account some recent developments in Nevanlinna theory and several complex variables. The treatment is differential geometric throughout, and assumes no previous acquaintance with the classical theory of Nevanlinna. The main emphasis is on holomorphic curves defined over Riemann surfaces, which admit a harmonic exhaustion, and the main theorems of the subject are proved for such surfaces. The author discusses several directions for further research.

Analytic functions. --- Functions, Meromorphic. --- Value distribution theory. --- Meromorphic functions --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Functions of complex variables --- Series, Taylor's --- Distribution of values theory --- Functions, Entire --- Functions, Meromorphic --- Addition. --- Algebraic curve. --- Algebraic number. --- Atlas (topology). --- Binomial coefficient. --- Cauchy–Riemann equations. --- Compact Riemann surface. --- Compact space. --- Complex manifold. --- Complex projective space. --- Computation. --- Continuous function (set theory). --- Covariant derivative. --- Critical value. --- Curvature form. --- Diagram (category theory). --- Differential form. --- Differential geometry of surfaces. --- Differential geometry. --- Dimension. --- Divisor. --- Essential singularity. --- Euler characteristic. --- Existential quantification. --- Fiber bundle. --- Gaussian curvature. --- Geodesic curvature. --- Geometry. --- Grassmannian. --- Harmonic function. --- Hermann Weyl. --- Hermitian manifold. --- Holomorphic function. --- Homology (mathematics). --- Hyperbolic manifold. --- Hyperplane. --- Hypersurface. --- Improper integral. --- Intersection number (graph theory). --- Isometry. --- Line integral. --- Manifold. --- Meromorphic function. --- Minimal surface. --- Nevanlinna theory. --- One-form. --- Open problem. --- Open set. --- Orthogonal complement. --- Parameter. --- Picard theorem. --- Product metric. --- Q.E.D. --- Remainder. --- Riemann sphere. --- Riemann surface. --- Smoothness. --- Special case. --- Submanifold. --- Subset. --- Tangent space. --- Tangent. --- Theorem. --- Three-dimensional space (mathematics). --- Unit circle. --- Unit vector. --- Vector field. --- Volume element. --- Volume form. --- Fonctions de plusieurs variables complexes