Hiya mathwonk
I was away for a few days, came back and settled down to sort out the relations
between curvature and the several kinds of derivatives on amanifold; and
I'm still going.
Anyway, my apology for imprecision above. I think the following is correct,
though I can't prove it explicitly:
(1) To every closed curve on a manifold, integration of the Riemann tensor
(over the area of any surface bounded by the curve) assigns an isometric
transformation of tangent bases (at the points of the curve).
(2) At anyone point, to any tangent bivector the Riemann tensor assigns
a generator of the isometry group: an "infinitesimal transformation", if you like.
So, although it is correct to say the Riemann tensor is a (1,1)-tensor-valued
2-form, it may be more revealing to say that it's a 2-form which maps the
tangent bivectors to the Lie algebra of the isometry group.
Now the Gauss Map is a construction I haven't seen for years, if at all. It
must display some aspect of the Riemann curvature in general - but I'm not
sure whether the parallel transport business leads to a simple way of doing
it.
For a 2D surface, the Gauss map is the unit sphere in 3D. For what it's worth,
I've found in
Exact Solutions of Einstein's Field Equations
Ed. E. Schmutzer
Cambridge University Press, 1980
the result
R_a_b_c_d = K (g_a_c g_b_d - g_a_d g_b_c)
where K is the Gaussian curvature, the Riemann tensor having only one
independent component. That doesn't get us terribly far.
The real key is the concept of the connection, which does have a direct
relation to the Gauss map, although I'm having a hard time making it
explicit. I think the connection is essentially the differential of
the Gauss map, but do treat the following with caution.
Explanations of the Gauss map that I've seen, especially:
http://www.mathwright.com/librarya/ccurves/ccurves4.htm
present it from the viewpoint of embedding a curve in R^2 or surface in
R^3.That makes a normal to the n-manifold in question a vector in
R^{n+1}, a sensible way to look at it. However from a
completely intrinsic point of view, I would characterise the "normal"
as the manifold "surface element": i.e. for any tangent basis
e_i, i=1,...,n, it is the wedge product of all the
unit tangents
e_1 \wedge e_2 \wedge ... \wedge e_n
Remove anyone term from this and apply Hodge duality to what remains. We get
the same term again, maybe up to a factor of -1.
Take the exterior derivative of the complete product, and we get a series of
terms
e_1 \wedge ... \wedge de_k \wedge ... e_n
each expressible via the Hodge duality as
<e_k, de_k>
That series is what the differential of the Gauss map would be.
de_k is the complete differential of e_k; the derivative
of e_k in the direction of e_l is
<e_l, de_k>
Despite appearances, this is a vector; and its component in the
de_m direction is
<e_m, <e_l, de_k>>
The set {de_k} *is what's called the connection, and the above
for all k,l,m are its components. In relativity work, using index notation,
the connection is written as the Christoffel symbol
\Gamma^k_{lm}
Now the point of all this is that formally we have taken the tangent basis
{e_i} and differentiated it. That's something that it makes
sense to do, only *if we have a function assigning one tangent basis to each
point of the manifold, and this function is smooth, i.e. actually has a
differential. That is actually the case, for example, when we have a
parametric curve in 2D andcan define its Serret-Frenet basis from its
velocity and acceleration vectors; or for another example, the Darboux
basis for a surface. This concept of a "frame field" and its differential
is generalized to the Cartan formalism of "the moving frame".
Simply, de_k is an element of the Lie algebra of the Lie group
which preserves the metric: the connected isometry group SO(n). In other
words de_k tells us what happens to e_k *as we move
across the manifold, by giving an infinitesimal transformation matrix <br />
<e_m, <e_l, de_k>> * for each direction e_l.
This is the mechanism which "rotates the spear". I had better try and make
this precise. To do so, I'm going to set out the whole geometric intuition as
best I can; so bear with me on what may seem a digression.
In many developments of differential geometry, we stay within a completely
intrinsic picture of a given manifold; we do not treat it by giving it an
embedding in a higher R^n. That will be the case here.
Consider a 2D surface, smooth but irregularly shaped; say a bowling pin. We
are concerned with what happens to a tangent vector ("the spear") as we move
it around the surface.
Obviously the spear _could_ wobble all over the place; it could be the tangent
vector to any curve we care to draw. However there is a natural way to move
it, so that it has no local change of direction: this is called "parallel
transport".
Let's look at a spear on a flat surface with a cartesian coordinate system.
Move it in a straight line. We will know its direction is unchanged, that is *
it is parallel to its former orientation, if its components are unchanged.
That defines parallel transport on a flat surface. But can we define the
equivalent on a bowling pin?
Yes we can: by transporting it along a geodesic curve - which is a locally
flat piece of the bowling pin's surface.
Take a flat sheet of paper and roll it into a cylinder. Despite having two
edges identified, it is still intrinsically flat, and the component criterion
for parallel transport still holds.
OK? Then take a long strip of paper, and roll it up like a roll of sticky
tape. If we printed cartesian coordinates with one axis down the centre, the
axis would overlay itself, right through the roll.
Lay the sticky tape down around a cylinder, perpendicular to the axis: it
overlays itself. Lay it down at an angle and it comes back parallel to
itself. This shows that the cylinder is flat.
Lay it on a sphere, and it will follow a great circle and come back to overlay
itself. It will crinkle a little at the edges, but we can make the crinkling
as small as we like, by using a narrower tape. Try to make it follow a line
of constant latitude at 30 degrees though, and it can't be done without
constantly bending and crinkling it to fit. Our sticky tape is a portable
standard of local parallelism: it is locally flat. It follows a geodesic
curve; and if you want to find the geodesic curve on a bowling pin, in a
given direction and through a given point, just pull out the sticky tape and
see where it goes.
But here's the crucial thing. On a curved surface, you cannot generally get
the sticky tape to stay parallel to a set of coordinate lines, no matter what
coordinate system you draw on it. If you start laying tape on the sphere
along the 30-degree-north line, the tape will diverge as it follows its own
great circle. The tangent bases along the tape and the tangent bases defined
by the spherical coordinates will relatively rotate. And the rate of
rotation, expressed in degrees per inch, or as a skew-symmetric
rotation-group generator matrix, is given by the connection (contracted with
the direction of the tape).
Just to align this with standard notation, the connection is often given the
symbol
\nabla
The connection along a given tangent, defined by a vector u is
\nabla_u
The infinitesimal change that makes to a vector v is
\nabla_u v
The component of that change measured by the 1-form w is
<w, \nabla_u v>
and the component of the change, when the vectors are the unit tangents and
1-forms defined by a coordinate system, is the Christoffel component
\Gamma^c_{ab} = <w^c, \nabla_(e_b) e_a>
Now out of all this, we want a definition of the curvature of the manifold at
a point. We're interested in the geometric invariants. For this purpose we
don't care what happens to a particular vector, let alone the components of
its variation. What counts is the connection in a given direction:
\nabla_u
This measures the rate at which the coordinate basis vectors are rotating with
respect to parallel transport. The connection in general,
\nabla, gives the rate of rotation corresponding to any rate of
movement across the manifold.
Now what we really want to know is the amount of rotation that happens around
a closed path, in the small limit. We choose as a path the circumference of
the area defined by two tangent vectors u and v. (Or strictly speaking, the
limit of the area defined by orbits tangent to u and v at their
intersections, as the lengths of the orbits approach zero.)
I'm going to quote the result, rather than provide the proof I read in Misner,
Thorne & Wheeler's _Gravitation_, which has some subtle features. Maybe there
is a neater proof around.
Assuming we have zero for the commutator \left u,v \right, which is
the case for coordinate line tangents, the net rotation per unit area is
\nabla_v \nabla_u - \nabla_u \nabla_v
That is the definition of the Riemann tensor.
And it's also as far as I've got. For an n-manifold, these rotations are in
SO(n). I'm certain there must be a map, based on these, which takes the
complete volume elements e_1 \wedge e_2 \wedge ... \wedge e_n to
the surface of the unit sphere in *R^{n+1}... i.e. the Gauss
map ... but I haven't seen it yet.
Thanks for the prompt. More anon, I hope.