The mathematical formalism of a smooth manifold
is a framework for
describing spaces that, locally, seem to be linear spaces, although they aren't
necessarily flat on larger scales – just as the surface of a huge ball
resembles a two-dimensional plane to an ant crawling on it, or the surface of
planet Earth might seem to be a crinkly variation around a flat horizontal plane
to people who have not travelled thousands of miles (in diverse directions)
while making careful geometric measurements.
One of the most important changes general
relativity forced in physical theory was the transition from modelling
space-time as a flat (vector) space to modelling it as a curved space. The
former can be equipped with a notion of distance that makes
it globally Euclidean (a formal
description of flat space): the latter can be given a notion of distance that
makes it locally Euclidean, but not in large
regions. (General relativity, on the other hand, equips space-time with
a Minkowskian metric rather than a Euclidean one; which is why the
fore-going says can be given; but that's a topic for another page.) The
mathematical formalism for describing curved spaces is the smooth
manifold (and doesn't depend on your notion of distance).
When primitive folk interpreted the Earth as (varying crinklesomely about) a flat plane, they were adopting a simple model that worked reasonably well; just so, physicists before Einstein modelled space as Euclidean because that was simple and gave answers that accorded with reality to within their ability to measure. In principle, in both cases, folk could have used as model a surface that looks flat locally but wobbles and curves on larger scales; had the world, or space, been flat they would have found this as a property of the surface they needed for their model, by measuring the wobbles and curvature and finding them to be zero; it would then have made sense to switch to the simpler model using a linear space – a particularly easy special case of the smooth manifold.
The theory of smooth manifolds builds up several layers of structure; on a
foundation of a topological manifold
(itself the top of several layers of
structure I'm glossing over), one imposes a smooth atlas
, which
identifies the charts, a.k.a. systems of co-ordinates – among which all
change-of-coordinate transformations are smooth – it is legitimate to
use. Given this, it is possible to build mutually dual vector spaces, at each
point, of tangents and gradients, hence spaces of all tensor-algebraic
combinations of these and spaces of linear maps among such spaces. Each recipe
for constructing a space from gradients and tangents in this way is termed a
tensor rank.
Although each such space is at each point, so each point's space of tensors of any given rank is distinct from any other point's space of that rank, it is possible (although I don't yet adequately address this) to specify continuity of variation of a function which, for some given rank, at each point, yields a tensor of the given rank at that point. It is even possible to specify what a differential operator is; but many differential operators are possible and there is nothing about a smooth manifold, per se, that gives us any reason to prefer one of these over any other. All of this is intrinsic to the smooth manifold without any regard to geometry.
When we come to add a geometry, the fore-going intrinsic structure of the smooth manifold encodes it as a symmetric tensor of type gradient times gradient; this tensor is called the metric. Each gradient factor intrinsically wants to contract with a tangent vector to produce a scalar, so the metric can eat two tangent vectors and produce a number, the inner product of the two tangents; as such, it's meaningful to ask whether it's symmetric (i.e. the order in which we feed it the two tangents doesn't matter); and usual kinds of geometry are indeed encoded by symmetric ones. The metric can equally be construed, by feeding it only one of the tangents, as a linear map from tangents to gradients – one gradient factor in the tensor eats the tangent and we're left with a scalar times the other – and, as such, we can ask whether it's invertible; for interesting geometry, it is. Its inverse is then a map from gradients to tangents – and, if you were taught that grad yields a (displacement-like; i.e. it looks and feels a lot like a straight line path from one point to another) vector derivative of a scalar function of position, this is because the intrinsic gradient of the scalar function can be fed through the inverse of the metric to obtain a tangent (which is as similar as you can get, on a smooth manifold, to a displacement while still being able to add and scale).
It turns out that – given a geometry encoded as such a (symmetric and
invertible) tensor field – among the broad class of possible differential
operators, there is exactly one, entirely determined by the metric, which
considers the metric constant
– i.e. its derivative of the metric
is zero. So the metric determines not only our notion of length but also our
notions of constancy and rate of variation. The differential operator, in turn,
can be used to obtain a characterization of the manifold's curvature.
A smooth manifold is a topological space with an open cover in which the
covering neighbourhoods are all smoothly isomorphic to one another. One
generally defines smoothly
for these purposes in terms of
a smooth atlas. As time goes by, I'll doubtless
fill that definition in with hyper-links to explain the jargon it contains. For
the present, what matters is that we obtain a notion of smoothness for mappings
to and from the manifold; notably, scalar functions of position (mappings from
the manifold to {scalars}) and paths, a.k.a. trajectories (mappings from
{scalars} to our manifold). From this we
can obtain a full tensor bundle for the
smooth manifold and a notion of smoothness
for tensor fields on the manifold.
Given the tensor bundle, we are in a position to describe a Riemannian or pseudo-Riemannian geometry on the manifold. This suffices to allow us the illusion of a Euclidean or Minkowskian view of all sufficiently small regions of the smooth manifold. The deviations from ideal (Euclidean or Minkowskian) geometry in insufficiently small regions can then be understood in terms of curvature, for the meaning of which we must look at differential operators on the manifold.
We will in any case need differential operators on our manifold if we are to build anything like our usual physical theories: we must be able to differentiate arbitrary smooth tensor fields (and the relevant derivatives must possess at least some sensible properties). This, in fact, leaves a fair amount of freedom to chose differential operators: however, if we insist on a differential operator which considers our metric (a tensor encoding the geometry) to be constant, this suffices to specify our differential operator uniquely (and there is such a differential operator).
A differential operator is a special case of
a Leibniz operator: it can be shown
that any Leibniz operator which annihilates all scalar fields acts as a linear
map on each rank of the tensor bundle: and that the antisymmetric self-product
of any Leibniz operator is a Leibniz operator (with twice the rank, of
course). Given that one of the sensible properties
referred to above is
that our chosen differential operator's square must act symmetrically on all
scalar fields, we now have a linear map describing its antisymmetric
self-product's action on gradients: this is known as
the Riemann tensor; it encodes the
curvature of our manifold. A contraction applied to the (fourth-rank) Riemann
tensor delivers the (second-rank) Ricci
tensor, which General Relativity uses to characterize the curvature of
space-time.
Written by Eddy.