Differential Geometry And Riemannian Manifolds

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Friedrich Sauvigny (Brandenburg University Of Technology Cottbus-senftenberg, Germany)

Differential Geometry And Riemannian Manifolds

Friedrich Sauvigny (Brandenburg University Of Technology Cottbus-senftenberg, Germany)

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English

World Scientific Publishing Co Pte Ltd
26 October 2025

Functional analysis & transforms; Differential & Riemannian geometry

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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.

By: Friedrich Sauvigny (Brandenburg University Of Technology Cottbus-senftenberg Germany)
Imprint: World Scientific Publishing Co Pte Ltd
Country of Publication: Singapore
ISBN: 9789819816163
ISBN 10: 9819816165
Pages: 376
Publication Date: 26 October 2025
Audience: College/higher education , Professional and scholarly , Primary , Undergraduate
Format: Hardback
Publisher's Status: Forthcoming