Ivancevic_Applied-Diff-Geom

Introduction 3Fig. 1.1 The four charts each map part of the circle to an open interval, and togethercover the whole circle.The surfaces of a sphere 3 and a torus 4 are examples of 2D manifolds.3 The surface of the sphere S 2 can be treated in almost the same way as the circle S 1 .It can be viewed as a subset of R 3 , defined by: S = {(x, y, z) ∈ R 3 |x 2 +y 2 +z 2 = 1}. Thesphere is 2D, so each chart will map part of the sphere to an open subset of R 2 . Considerthe northern hemisphere, which is the part with positive z coordinate. The function χdefined by χ(x, y, z) = (x, y), maps the northern hemisphere to the open unit disc byprojecting it on the (x, y)−plane. A similar chart exists for the southern hemisphere.Together with two charts projecting on the (x, z)−plane and two charts projecting on the(y, z)−plane, an atlas of six charts is obtained which covers the entire sphere. This canbe easily generalized to an nD sphere S n = {(x 1 , x 2 , ..., x n) ∈ R n |x 2 1 + x2 2 + ... + x2 n = 1}.An n−sphere S n can be also constructed by gluing together two copies of R n . Thetransition map between them is defined as R n \ {0} → R n \ {0} : x ↦→ x/‖x‖ 2 . Thisfunction is its own inverse, so it can be used in both directions. As the transition mapis a (C ∞ )−smooth function, this atlas defines a smooth manifold.4 A torus (pl. tori), denoted by T 2 , is a doughnut–shaped surface of revolution generatedby revolving a circle about an axis coplanar with the circle. The sphere S 2 is aspecial case of the torus obtained when the axis of rotation is a diameter of the circle. Ifthe axis of rotation does not intersect the circle, the torus has a hole in the middle andresembles a ring doughnut, a hula hoop or an inflated tire. The other case, when theaxis of rotation is a chord of the circle, produces a sort of squashed sphere resembling around cushion.A torus can be defined parametrically by:x(u, v) = (R + r cos v) cos u, y(u, v) = (R + r cos v) sin u, z(u, v) = r sin v,where u, v ∈ [0, 2π], R is the distance from the center of the tube to the center of thetorus, and r is the radius of the tube. According to a broader definition, the generatorof a torus need not be a circle but could also be an ellipse or any other conic section.Topologically, a torus is a closed surface defined as product of two circles: T 2 = S 1 ×S 1 . The surface described above, given the relative topology from R 3 , is homeomorphicto a topological torus as long as it does not intersect its own axis.One can easily generalize the torus to arbitrary dimensions. An n−torus T n is definedas a product of n circles: T n = S 1 × S 1 × · · · × S 1 . Equivalently, the n−torus isobtained from the n−cube (the R n −generalization of the ordinary cube in R 3 ) by gluing