Since the mathematical formalism is quite involved, it seems like a good idea to include an illustration of what a smooth manifold is like. By chosing one that we generally have a good intuitive feel for, I hope I can illuminate the meaning of the formalism. The sphere is an intuitively staightforward enough smooth manifold that it presents itself as a natural candidate: furthermore, the mathematical formalisms needed to describe it in terms of flat space are fairly straightforward; and the same formalisms suffice in all dimensions.
Now one of the details that's going to matter is curvature and a smooth manifold can have differing curvatures in differing directions; so rather than actually look at a sphere (which doesn't exhibit this feature) I'll look at an ellipsoid - the result of scaling a sphere by differing amounts in various orthogonal directions. This will then have the requisite diversity of curvatures, packaged in a form that's intuitively tractable.
So first let's suppose we have a vector space, V, of dimension 1+dim (so that the surface of its sphere has dimension dim) in which we have a positive-definite metric (dual(V): h |V), giving us our conception of length: h(x,x) is the square of the length of x. We then have a second positive-definite quadratic form (dual(V): H |V) which we use to define our ellipsoid
Linear algebra in V will let us chose a basis (V: b |1+dim) with dual (dual(V): q |1+dim) - so q(j)·b(i) is 1 if i=j, otherwise 0 - for which h = sum(: q(i)×q(i) ←i |1+dim) and H = sum(: q(i)/r(i)×q(i)/r(i) ←i |1+dim) with ({scalars}: r |1+dim) being the radii of our ellipsoid in the various directions; each ±r(i).b(i) is a point on the ellipsoid.
From this description of our manifold, S, embedded in a flat space, V, we
now need to obtain a description of the manifold itself, without reference to
any embedding. To do that we need to identify the metric on S as a function of
position within S. To describe positions within S we'll need an atlas of S, but
this is easily obtained from our co-ordinates in V; for any j in 1+dim define
(dim: J :1+dim) by J(i) = i if i < j, J(i) = i-1 if i > j; then use (: (:
x·q(i) ← J(i) :dim) ←x :) as a chart on each of {x in S:
x·q(j) >k} and {x in S: x·q(j) <k} for some positive k much
smaller than r(j). These charts, for the various j in 1+dim, fully cover S
(each omits a neighbourhood of the equator
for its co-ordinate; but
nowhere is close to all equators, e.g. if each k is < r(j) / √(1+dim);
any x in V for which x·q(j) < r(j) / √(1+dim) has H(x,x) < 1
so isn't in S). Thus we can use co-ordinates in V as co-ordinates in S for all
practical purposes.
Now the metric on S is induced from (dual(V): h |V) = sum(: q(i)&tensor;q(i) ←i |1+dim), save that we need to express it in terms of gradient fields on S, not co-vectors in V.
Written by Eddy.