In algebra we added arithmetic properties to sets cumulatively until we had built enough structure to arrive at the real numbers. We then went further by considering generalizations of vectors. In geometry, we build up to a surface in three-dimensional space, and then go further by considering generalizations of tangent vectors to a surface.
In this section we will give a preview of the basic ideas, with the implicit assumption that the reader already has some familiarity with objects and relations that will only be defined later.
| Added structure | Resulting features | |
|---|---|---|
| Topological space | Open sets | Connectedness, holes |
| Hausdorff space | Disjoint neighborhoods | Separation between points |
| Metric space | Metric | Distances between points |
| Topological manifold | Cartesian charts | Coordinates, dimension |
| Differentiable manifold | Differentiable structure | Tangent vectors, calculus |
| Riemannian manifold | Riemannian metric | Length, angles, volumes |
| Euclidean surface | Embedding in \({\mathbb{R}^{3}}\) | Relation to a higher space |
Notes: There are many more messy details here than with algebra. These will be covered in the following chapters.
\({\mathbb{R}^{n}}\) denotes the Euclidean space of dimension \({n}\), i.e. the manifold of points which are \({n}\)-tuples of real numbers \({x^{\mu}}\) with the Euclidean metric; similarly, \({\mathbb{R}^{r,s}}\) denotes pseudo-Euclidean space, the same manifold with the pseudo-Euclidean metric of signature \({(r,s)}\), while \({\mathbb{C}^{n}}\) denotes a complex manifold. This notation applies even if some of the structure is ignored, e.g. if the context is topological spaces, then the metric on \({\mathbb{R}^{n}}\) is ignored.
| Δ A possible source of confusion is the overloading of the above notation for vector spaces, and the status of the origin as a special point, which is left ambiguous. Euclidean spaces are sometimes denoted \({\mathbb{E}^{n}}\) to distinguish them from the vector space \({\mathbb{R}^{n}}\), but we do not follow this convention since it is less common, the distinction is usually clear from context, and it leaves the status of the origin ambiguous. |