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mathematics that studies the properties of spatial objects from their inherent connectivity while ignoring the detailed form. The examples in Figure 04 shows that the 2-dimensional objects are separated into different class (space) according to the number of holes in them. Objects in each space have the same topology as long as they can be moulded into each other by deformation, twisting and stretching but no discontinuous operation such as cutting. Topology in higher dimension is more complicated than the 2-dimensional example, but the general idea remains the same. The classification of topological equivalent objects depends on the Euler characteristic = V - E + F, where (for 1 or 2 dimensional object) the number of vertices V = # of (n+1) or more edges meeting at a point + # of end points (n is the dimension of the object), the number of edges E = # of intersections of 2 faces or more + # of boundaries, and F is the number of faces. Table 01 shows some of the geometric objects | |
Figure 04 Topology |
together with the Euler characteristic. The objects with the same Euler characteristic belong to the same topological class even though they could look very different. |
1. For 1 complex dimensional manifold, there is just one Chern class, the first Chern class, which equals the Euler characteristic. In Figure 05, the complex 1-D sphere has 2 spots with zero net flow implying a first Chern class = Euler characteristic = 2, while the torus has no such spots and has first Chern class = Euler characteristic = 0. 2. The 2 complex dimensional manifold has a first and a second Chern class. The first one assigns an integer coefficients to subspaces (the 1-D manifold within the 2-D manifold), while the second one (the last one) always equals to the Euler characteristic. 3. The n complex dimensional manifold has a first and a second, ... n^{th} Chern classes. The | |
Figure 05 1-D Chern Classes |
first one assigns an integer coefficients to subspaces (the 1-D manifold within the n-D manifold), while the second one assigns another number to subspaces (the 2-D manifold within the n-D manifold), ..., the last one always equals to the Euler characteristic. |
K = |d(dy/dx)/dx| / [1+(dy/dx)^{2}]^{3/2} ---------- (1) while the square of an element of an arc on the curve is : ds^{2} = [1+(dy/dx)^{2}] dx^{2} ---------- (2) The factor in front of dx^{2} is the 1 dimensional metric tensor g_{11} for a smooth manifold. An Euclidean example is a straight line described by the formula : y = ax + b ---------- (3) which produces K = 0, and ds^{2} = (1+a^{2}) dx^{2}. For a horizontal line such as the one shown in Figure 06, a = 0 and ds = dx. A circle provides an 1 dimensional non-Euclidean example, it is expressed by the formula : x^{2} + y^{2} = r^{2} ---------- (4) which yields K = 1/r ---------- (5), | |
Figure 06 Curvature of 1-D Object [view large image] |
and ds^{2} = r^{2}/(r^{2} - x^{2}) dx^{2} ---------- (6).
Note that there are two singularities at x = _{}r. However, integration from x=-r to x=r returns a finite arc length of s = r. |
the hyperbolic paraboloid surface. According to the Gauss-Bonnet Theorem the sum of the Gaussian curvatures over the whole surface equals to 2 multiplied the Euler characteristic , i.e. : _{}_{A} K dA = 2 ---------- (7) This theorem means that the overall curvature of the surface is fixed. Bending and pulling the surface may alter the curvature at every point, but these changes all cancel each other out. A trivial example is the surface of a sphere with unit radius (see insert in Figure 07). Its Gaussian curvature is 1 everywhere and | |
Figure 07 Curvature of 2-D Object [view large image] |
the total surface area is 4. Thus = 2, which is the same as calculated previously by another method. See pseudosphere for an example of constant negative curvature. |
where | ^{,} | R = g^{ik}R_{ik} is the scalar curvature, g_{ik} is the metric tensor in the |
is the Christoffel symbol, and T_{ik} is the energy-momentum tensor, the indices |
| |
Figure 08 Conformal Mapping [view large image] |
3. Parallel transport is a process of moving vectors or spinors along a path on a manifold that keeps the lengths as well as the angles between any two vectors unchanged (see Figure 09a for moving a vector on spherical surface by maintaining a southerly direction and tangential to the surface. if the direction changes at the end of a closed path, then there is curvature over the path). 4. Holonomy Group contains all the elements, which are closed paths formed by parallel transport, e.g., the path 1, 2, 3 in Figure 09. | |
Figure 09 Parallel Transport Symmetry [view large image] |
5. For a Ricci flat manifold, a vector or spinor retains its orientation after moving through a closed path. Such spinor is called "covariantly constant spinor" and is a member of the SU(3) (3 complex or 6 real dimensions) holonomy group. |
Figure 10 Calabi-Yau Manifold [view large image] |
of the 2 dimensional cross-section from a 6 dimensional Calabi-Yau manifold. |