Elements of Differential Geometry

In this chapter we begin with an acquaintance of basic notions of topology. This field of geometry studies topological properties of figures, that is, the properties which are preserved under deformations that do not tear or glue together parts of the figure. Thus, the circumference, the boundary of a square, and the knot in Figure 1 have the same topological properties, while the sphere and the surface of a bread-ring have different topological properties. The first part of this chapter (Sections 1–7) is devoted to these questions. As a rule, some geometrical objects depending on parameters appear at the very first steps of formalization of various concrete applied problems. For example, the set of all positions of a pendulum of constant length in space is a sphere; in different terms, the set of all positions (the configuration space) of this mechanical system is parametrized by points of the two-dimensional sphere. Another example is the set of all straight lines in the plane. This space appears in geometrical optics, and its structure is more complicated than the one of the sphere. The two spaces have a common property: every point has a neighborhood topologically equivalent to a disk. The spaces with this property are called manifolds. The second part of this chapter (Sections 8–14) contains an introduction to the geometry of smooth manifolds. To construct a geometry in which one can define notions as in the usual Euclidean geometry, such as length, angle, volume, straight line and movement, one needs an additional structure, called the Riemannian metric. This leads to the theory of Riemannian manifolds. An introduction hereto is given in Sections 15–17.