Ivancevic_Applied-Diff-Geom

Chapter 3Applied Manifold Geometry3.1 IntroductionAlbert Einstein once said: “Nature is simple only when analyzed locally.Why? Because, locally any system appears to be linear, and therefore fullypredictable and controllable. Geometrical elaboration of this fundamentalidea has produced the concept of a manifold, a topological space whichlocally looks like Euclidean R n −spaces, but globally can be totally different.In addition, to be able to use calculus on our manifolds, in much thesame way as in ordinary R n −spaces, the manifolds need to be smooth (i.e.,differentiable as many times as required, technically denoted by C k ).Consider a classical example, comparing a surface of an apple with aEuclidean plane. A small neighborhood of every point on the surface ofan apple (excluding its stem) looks like a Euclidean plane (denoted byR 2 ), with its local geodesics appearing like straight lines. In other words,a smooth surface is locally topologically equivalent to the Euclidean plane.This same concept of nonlinear geometry holds in any dimension. If dimensionis high, we are dealing with complex systems. Therefore, whilecontinuous–time linear systems live in Euclidean R n −spaces, continuous–time complex systems live in nD smooth manifolds, usually denoted byM.Finally, note that there are two dynamical paradigms of smooth manifolds:(i) Einstein’s 4D space–time manifold, historically the first one, and(ii) nD configuration manifold, which is the modern geometrical concept.As the Einstein space–time manifold is both simpler to comprehend andconsequently much more elaborated, we start our geometrical machinerywith it, keeping in mind that the same fundamental dynamics holds for all137