Wednesday, April 6, 2011

Manifolds: Curvature and Construction

This is part of a series on Manifolds. For more information about subjects mentioned, see The Complete Manifolds Series.

On the previous post, it became possible to define unique points on a manifold by the use of coordinates. Consequently, the connecting of these points forms lines.

However, if one insists on staying on the surface of a manifold, the lines will not necessarily be straight. Take the sphere for example. Any lines drawn on the surface will inevitably be curved. But what does curvature mean? And how does one choose the single line that connects two points? The answer is a simple one.

On a surface, the line between two points is the shortest curve that connects them. On a plane, these lines are straight. On a sphere, the lines are arcs, part of great circles whose radius is that of the sphere itself (see below). On a general manifold, the shortest path between two points is known as a geodesic.



Each circle above is defined by a plane passing through a sphere. If the plane passes through the center of the sphere, the resulting circle is a great circle. Otherwise, it is not. As a result, all longitude lines are great circles, but, among latitude lines, only the Equator is.

After identifying the lines on a sphere, one can continue on to produce geometric figures. Take the triangle for example. It is defined as the (minor) area of the sphere contained by three geodesics on the surface. With a little thought, however, triangles on this surface can be identified to have angular measures greater than 180º. Begin at the north pole of the sphere. The 0º and 90º W longitude lines both emanate from this point, and are by definition, perpendicular. These two lines both intersect the Equator at right angles. (longitude and latitude lines are always perpendicular) These are all clearly geodesics, and each pair contains a right angle. Consequently, the total angular measure of the triangles is 270º! The variation from 180º of the angular measure of a triangle on a manifold is known as the angular excess. A basic discussion of angular excess and resulting curvature can be found here.

In the previous post in this series, the construction of manifolds through coordinate systems was discussed. Although some interesting cases are found through this method, it only produces a small number of manifolds. Another method of construction is through the connection of boundaries. A manifold is said to have a boundary if it has an edge.

For example, the circle does not have a boundary, but the disc does, and the sphere does not have a boundary, but the ball does. Also, the plane is infinite in all directions, and therefore doesn't have a boundary.



However, by connecting the boundaries of two discs, as shown above, one obtains a sphere. (remember that bending and stretching is allowed in a mapping, but no breaking) Also, it is clear that connecting two manifolds preserves their dimension, but turns two discs (manifolds with boundary) to a single sphere (a manifold without boundary).

For two-dimensional manifolds, cuts and folds can be made on the surface to reduce the manifold to a polygon with an even number of sides. This polygon is known as the fundamental polygon.

As an example, consider the torus. By making two cuts, it can reduced to a rectangle.



In the first step, the cut goes through the torus and creates two ends. The resulting figure can be straightened out into a cylinder.



A second cut is made parallel to the surface to the cylinder along the plane indicated. The cylinder can then be unrolled into a rectangle. The result of this process is summarized in the following diagram.



The fundamental polygon for the torus. It has four sides, and can therefore be stretched into a square continuously, as it is shown here. The orientation and lettering of the sides indicates which way the boundaries of the square are connected to produce the manifold. The sides with the same letter are connected to each other so in such a way that the arrows face the same direction. Here the top and bottom are connected, and then the left and right. It is easy to see that this is the reverse of the process that we used to decompose the torus.

The same process can be applied to the sphere. The fundamental polygon for the sphere is again a square, but it is connected in a rather different way.



The above cut is made before "flattening" the sphere into the plane figure below.



This fundamental polygon has the same structure as that of the torus, but is connected differently. The points of each arrow must line up in the end result. With a square piece of paper, one only has to fold diagonally to line up the indicated arrows. The figure is then "inflated" to produce the sphere.

The fundamental polygons can be written as follows. Going in a clockwise direction from the upper left corner, the fundamental polygon for the torus is written ABA*B* and, for the sphere, ABB*A* (the * in each case represents an arrow pointing in the counterclockwise direction).

Next, it is possible to extend this system to other two-manifolds, and even to other dimensions. A closer examination of these figures gives insight into curvature as well.

Sources:
http://whites-geometry-wiki.wikispaces.com, http://svr225.stepx.com:3388/sphere, http://en.wikipedia.org/wiki/Fundamental_polygon

2 comments:

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manoj singh said...

Thanks for sharing the information and I am here to share some general information about perpendicular lines that is, They are the lines that are at right angles (90°) to each other.
What is a Quadrilateral