Ivancevic_Applied-Diff-Geom

4 Applied Differential Geometry: A Modern IntroductionManifolds are important objects in mathematics, physics and control theory,because they allow more complicated structures to be expressed andunderstood in terms of the well–understood properties of simpler Euclideanspaces.The Cartesian product of manifolds is also a manifold (note that notevery manifold can be written as a product). The dimension of the productmanifold is the sum of the dimensions of its factors. Its topology is theproduct topology, and a Cartesian product of charts is a chart for the productmanifold. Thus, an atlas for the product manifold can be constructedusing atlases for its factors. If these atlases define a differential structureon the factors, the corresponding atlas defines a differential structure onthe product manifold. The same is true for any other structure definedon the factors. If one of the factors has a boundary, the product manifoldalso has a boundary. Cartesian products may be used to construct tori andcylinders, for example, as S 1 × S 1 and S 1 × [0, 1], respectively.Manifolds need not be connected (all in ‘one piece’): a pair of separatecircles is also a topological manifold(see below). Manifolds need not beclosed: a line segment without its ends is a manifold. Manifolds need notbe finite: a parabola is a topological manifold.Manifolds can be viewed using either extrinsic or intrinsic view. In theextrinsic view, usually used in geometry and topology of surfaces, an nDmanifold M is seen as embedded in an (n + 1)D Euclidean space R n+1 .Such a manifold is called a ‘codimension 1 space’. With this view it iseasy to use intuition from Euclidean spaces to define additional structure.For example, in a Euclidean space it is always clear whether a vector atsome point is tangential or normal to some surface through that point. Onthe other hand, the intrinsic view of an nD manifold M is an abstractway of considering M as a topological space by itself, without any need forsurrounding (n+1)D Euclidean space. This view is more flexible and thus itis usually used in high–dimensional mechanics and physics (where manifoldsused represent configuration and phase spaces of dynamical systems), canmake it harder to imagine what a tangent vector might be.Additional structures are often defined on manifolds. Examples of manifoldswith additional structure include:• differentiable (or, smooth manifolds, on which one can do calculus;the opposite faces together.An n−torus T n is an example of an nD compact manifold. It is also an importantexample of a Lie group (see below).