DIFFERENTIAL GEOMETRY AND RIEMANNIAN MANIFOLDS

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.

Contents:

• The Theory of Curves in ℝⁿ
• The Local Theory of Surfaces in ℝ³
• The Global Theory of Surfaces in ℝ³
• Geodesic and Harmonic Mappings in Riemannian Spaces
• The Global Theory for Riemannian Manifolds

Readership: For lectures on differential and Riemannian geometry aimed at graduate students in mathematics and physics, as well as for high school teachers seeking an advanced reference.