Geometry is concerned with the shapes (sizes, angles, etc.) of structures in a given domain. At bottom, it is dictated by the phenomenon of distance in that domain: this, in turn, implies notions of area, volume, etc. on sub-domains of higher dimension. So, in fact, this page is really about distance. (You can, however, find pages on things more usually viewed as geometry elsewhere among my pages.)
There are a variety of ways one may define distance. A standard axiomatic
approach to distance, for instance, would give it as a function taking two
points in one's space and returning a numeric answer subject to some
constraints: one commonly requires symmetry (though not in a
metric, which allows for a one-way system), non-negativity (with zero
distance meaning the two places are, in fact, the same place) and the triangle
law (the distance between two places is never more than the sum of their
distances from some third place). In relativistic discussions, however, the
notion of distance needs to cope with the squares of space-like and time-like
distances having opposite sign.
In a vector space, a very natural form of distance arises: any linear map from the vector space to its dual defines a product on vectors delivering a scalar value (the linear map eats up one vector to produce a covector; this, in turn, can eat up a second vector). This is said to be symmetric if the product of any two vectors is the same in whichever order they are used. It is said to be positive-definite if the product of any non-zero vector with itself is positive (it being easy to show that the product of any vector with the zero vector is zero) and positive semi-definite if the self-product of any vector is non-negative. We can then define the length of any vector to be the square root of the vector's self-product and the distance between any two points as being the length of the vector displacement between them. For a positive-definite symmetric linear map of this kind, this distance is then a distance in the axiomatic sense given above. The linear map from the vector space to its dual which defines such a product is called a metric on the vector space in question.
In a vector space with such a metric (assumed positive-definite and symmetric), one can extend the notion of distance between points to a notion of length of general curves in the space. The square root of the self-product, under the metric, of a curve's tangent vector provides a function which may be integrated along the curve to give a scalar result which, like the vector integral of the vector itself, does not depend on the parameterisation used to describe the curve. This integral coincides with the distance when the curve is the straight line path between two points and attains a global minimum there.
The same construction can, in fact, be generalised to cope with general metrics: one may, immediately, define the length of any curve whose tangent has positive self-product at all points. By allowing complex distances, we can extend this treatment to arbitrary curves provided that we always use the same sign of imaginary part when we take the square roots of negative values. Having departed into the complex world, we no longer have minimal length curves (because the complex numbers don't have the requisite notion of less than) but, instead, stationary or extremal length curves. Again, the straight lines provide curves of extremal length, as may readilly be shown.
When a notion of distance may be shown to be expressible as the distance induced by some metric, I shall describe it as metrisable. It is worthy of note that the distances defined by a metric depend only on the symmetric part of the metric (since it is used, in the construction of distance, only in self-products of tangents, in which only the metric's symmetric part can play a hand). Consequently, one generally works with this symmetric part and ignores any antisymmetric part. Hereafter, anything I refer to as a metric should be understood to be (implicitly) symmetric – though, some day, I must look into the guage invariance implied by our freedom to replace our metric with any other of which it is the symmetric part: what physical process does this reveal as its guage field ?
On a smooth manifold we do not have a vector displacement between points but
we do have trajectories and these do have tangent vectors. We can have a tensor
field on the manifold whose value at each point is a metric on the tangent
vector space. Consequently, we can define the path-length of trajectories with
respect to a metric, exactly as for a vector space (except that the metric can
no longer be regarded as
the same everywhere, at least not unless we construct our notion of
constancy specifically to
regard it as such).
A metriseable geometry on a smooth manifold is said to be pseudo-Reimannian: when the metric in question is positive-definite the geometry is described as Reimannian. It is not especially hard to demonstrate equivalence between being Reimannian and being apparently Euclidean with respect to some chart in sufficiently small neighbourhoods of every point on the manifold.Written by Eddy.