Schutz: GMoTP 2.0 - 2.3

Enter the main characters in his drama: Differential Manifolds and Tensors

We have seen that the possibility of defining continuous maps (or functions) is the key ingredient that makes topological spaces interesting. A differentiable manifold is a space on which differentiable functions can live. The physical space in which we live is thought to be such a space, and so are many other interesting mathematical objects, such as the space of solutions to differential equations. Fundamentally, a differentiable manifold is a space which looks locally like R^n. Each point in the manifold has an open neighborhood which has a continuous 1-1 map onto an open set of R^n for some n.

2.1 Definition of a Manifold

The key point here is that for a general manifold M, no single map from M->R^n will do. In general, multiple overlapping maps (an Atlas) will be needed, and for M to be a diffentiable manifold, those maps must overlap smoothly, permitting differentiable coordinate transformations in the overlap regions. The technical definition requires a bit of study for understanding. Helpful diagrams are provided.

2.2 The Sphere as a Manifold

The most familiar example of a manifold that isn't just R^n is the two Sphere S^2 - the set of points comprising the surface of a sphere. It is not possible to cover the surface of the sphere with a single map from S^2 -> R^2. Schutz notes that this fact follows from the topology of the 2-sphere, and is equally true topologically equivalent manifolds. Schutz looks at the sphere and illustrates some points about the nature of manifolds with diagrams and discussion.

2.3 Other Examples of Manifolds

Six examples of other manifolds: the set of rotations of a sphere, the Lorentz transformations of special relativity, the 6N positions and velocities of N particles (phase space), the space of dependent and independent variables of an equation, any vector space, a Lie Group - to be discussed in detail later.

Still no problem sets. Vacation is not quite over yet!