Lectures on Differential Geometry

This book is a translation of the Russian original that is based on lecture notes from a one-semester course for undergraduate students given by I. A. Taimanov in 1998. It consists of three parts. The first part is devoted to a study of curves and surfaces in Euclidean 2- and 3-space. The second part covers basics of smooth manifolds and Riemannian geometry, including basic properties of curvature, sectional curvature, the Levi-Cività connection and geodesics. Many interesting examples are given, including a trivial proof that a smooth manifold may be embedded to Rn for large n. Finally, the Lobachevski plane is described as a Riemannian manifold, whereas the Lie group PSL(2,R) is recognized as being the group of Lobachevski transformations. The third part covers more advanced materials. It begins with a proof that any two two-dimensional surfaces are locally conformally equivalent, followed by a description of a relation between umbilic points and the Hopf differential. The proof of the Hopf theorem (any topological sphere with constant mean curvature is isometric to the standard sphere) is given. Further topics included are: minimal surfaces, basics of Lie group theory and Lie algebra theory, basics of representation theory, including the proof of the Peter-Weyl theorem, and the definition of the Fourier transform on a group. In the last chapter, fundamental facts on Poisson and symplectic geometry are presented, leading to integrable Hamilton systems and Hamilton’s variation principle. The book is aimed at undergraduate students, as well as anyone who wants to learn basic facts from differential geometry.

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