Differential Geometry And Riemannian Manifolds
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€122.99
A01=Friedrich Sauvigny
Author_Friedrich Sauvigny
Bernstein's Theorem
Category=PBKF
Category=PBMP
Complete Riemannian Manifolds
Covariant Derivatives
Curve Theory
Energy Functional and Geodesics
eq_isMigrated=1
eq_nobargain
Fundamental Equations for Surfaces with Isothermal Parameters
Gauss-bonnet Formula for Conformally Parametrized Surfaces with Singular Boundaries
Geodesics in Riemannian Spaces
Hadamard-Cartan Theorem for Manifolds
Hopf's Theorem on Closed Surfaces of Constant Mean Curvature
Isoperimetric Problem in R3
Isoperimetric Problem in the Plane
Main Theorems in Surface Theory
The Exponential Map
Product details
- ISBN 9789819816163
- Publication Date: 04 Nov 2025
- Publisher: World Scientific Publishing Co Pte Ltd
- Publication City/Country: SG
- Product Form: Hardback
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This textbook focuses on the study of curves and surfaces, applying modern differential equation theory to geometric problems. By introducing isothermal parameters, it simplifies the fundamental equations of surface theory, leading to clear derivations of results like those of H Hopf and S Bernstein for surfaces of constant and vanishing mean curvature.Deviating from traditional approaches, the book first treats n-dimensional Riemannian spaces by a corresponding metric, then constructs Riemannian manifolds through transition conditions. The ultimate goal is to prove the Hadamard-Cartan theorem on the diffeomorphic character of the exponential mapping in Riemannian manifolds with nonpositive sectional curvature. By considering curves and surfaces in their optimal parametrization, the resulting ODEs and complex PDEs can be analytically solved, eliminating the need for intricate tensor calculus.The approach follows that of G Monge in his treatise L'Application de l'Analyse à la Géométrie, applying analytical techniques to geometric problems. Building on this foundation, the book uses modern theory of ODEs and PDEs to study the local and global results for curves and surfaces and their relevant curvatures.
