Differential and Complex Geometry: Origins, Abstractions and Embeddings
Springer, 01.08.2017 - 319 Seiten
Differential and complex geometry are two central areas of mathematics with a long and intertwined history. This book, the first to provide a unified historical perspective of both subjects, explores their origins and developments from the sixteenth to the twentieth century.
Providing a detailed examination of the seminal contributions to differential and complex geometry up to the twentieth-century embedding theorems, this monograph includes valuable excerpts from the original documents, including works of Descartes, Fermat, Newton, Euler, Huygens, Gauss, Riemann, Abel, and Nash.
Suitable for beginning graduate students interested in differential, algebraic or complex geometry, this book will also appeal to more experienced readers.
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Part II Differential and Projective Geometry in the Nineteenth Century
3 Projective Geometry
4 Gauss and Intrinsic Differential Geometry
5 Riemanns HigherDimensional Geometry
Part III Origins of Complex Geometry
9 Complex Analysis
10 Riemann Surfaces
11 Complex Geometry at the End of the Nineteenth Century
Part IV TwentiethCentury Embedding Theorems
12 Differentiable Manifolds
13 Riemannian Manifolds
14 Compact Complex Manifolds
15 Noncompact Complex Manifolds
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Abel Abel’s Abelian functions Abelian integrals algebraic curves algebraic topology analysis analytic Cauchy Chern classes coefficients cohomology groups compact complex manifold complex geometry complex manifold complex numbers complex plane complex variables consider coordinate chart coordinate system curvature defined denote Descartes developed differentiable manifold differential forms differential geometry dimension discuss divisor domain elliptic functions elliptic integrals embedding theorem Euclidean space Euler exact sequence finite follows formulated fundamental Gauss genus given Grauert harmonic hence Hermitian metric Hodge holomorphic functions holomorphic line bundle important inverse Jacobi Kähler manifold Kodaira Lemma linear mapping f mathematical mathematicians meromorphic function Nash’s neighborhood nineteenth century notation one-form open set paper polynomial problem projective geometry projective space proof published rational function real-analytic Riemann surface Riemann–Roch theorem Riemannian metric Sect sheaves solution Stein manifold submanifold subset surjective theory transition functions vector bundles vector space Weierstrass