!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
The "staircase" model of the blowup of the projective plane at the
origin (of a variety at a point),
as on the cover of Shafarevich II
looks A LOT like *torsion*!
[That is, just as an orbifold is like a manifold with singular curvature,
(er, curvatures that are delta functions)
a blowup is like a manifold/variety with singular *torsion*]
That is, go off in one direction,
rotate around a bit, come back along another
...and you'll have moved up the curves of the blowup!!
[blowup as "resolution" of torsion singularity?]
singularities of varieties / of maps
node is failure of embedding (pass to cover)
cusp (curvature singularity) is failure of immersion
blow-down sorta weird now...
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
re: torsion
I'm very confused
It looks to me like d\theta
gives a 1-form (with values in the trivial bundle)
that has torsion.
Note: it does not take values in the tangent bundle,
so you get torsion on a connection that isn't the tangent bundle
(and can't be)
...which contradicts what Silly rabbit said
Properly, d\theta (define on plane) gives a singular form;
it's gross at the origin.
You can flatten it out, either as r d\theta or r^2 d\theta,
and still have torsion
(concretely, go out along a radial path, go around a loop,
come back)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
pithily
$$R := \nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]}$$
$$R := [\nabla_X,\nabla_Y] - \nabla_{[X,Y]}$$
$$R := [[],\nabla]$$ (first do $\nabla$, then $[]$)
properly:
$$R := [[],\Delta^* \nabla]$$
So:
curvature is the non-commutativity of Lie derivative and covariant
derivative.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"Affine connection" sounds like
"connection on fiber bundle with structure group Aff"
...but apparently it's
"connection on tangent bundle"?
[confused!]
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
\chapter{Interpreting curvature}
Curvature and torsion of an affine connection are the infinitesimal holonomy;
curvature is the linear/rotational part, torsion is the translational part.
The connection $1$-form $\omega$ and the curvature $2$-form $\Omega$
represents holonomy (over a path or around the boundary of a surface).
[hmm...actually, should you think of integrating $\Omega$
over a $2$-cell/surface, or over a moving 2-frame (sorta; can fail to be a 2-frame),
namely velocity & acceleration
Cartan's method of moving frames]
for a Lie group, the connection form is aka the Maurer-Cartan form
[I'm told that torsion is rather subtle,
and that one needs Spencer cohomology (deep theory of higher order
differential equations)
to really understand what's going on.]
urk! I think an affine connection is a connection on the tangent bundle??
connection form is an infinitesimal notion;
parallel transport is rather *local*
parallel transport is the modern term for ``development''
Curvature is a (1,3)-tensor, and torsion is a (1,2)-tensor;
these are complex objects, and hence understandably hard to interpret.
[Here thinking of them as tensors is just confusing]
useful to think as so(n)-valued 2-form,
instead of just as tensor with some weird symmetries
Here is an interpretation in terms of holonomy (parallel transport).
These are properties of a *connection*, not a *metric*;
a metric picks out a distinguished connection
(the Levi-Civita connection),
but it's easiest to go via this intermediate step
They are the rotational and translational parts of infinitesimal holonomy
in a plane.
curvature = rotational,
torsion = translation
indeed, can see them together,
as an $\Aff TM$-valued 2-form.
Think of them as 2-forms,
taking values in End TM and TM, respectively
(operator-valued and vector-valued).
Now interpret basic elements of $\Lambda^2 TM$ as weighted planes:
thus 2-forms evaluate on planes.
(curvature tensor measures noncommutativity of the covariant derivative)
people often state it in terms of a parallelogram,
which is fine, but I find a "loop in the plane" a bit more suggestive
[sectional curvature = Gaussian curvature of \sigma-section at p;
however, this doesn't capture the intuition of the full curvature form;
it's just how much you turn in that plane
(i.e., the projection of the holonomy into the plane)]
[Ricci curvature is natural;
careful though: it's not the trace of the operator:
you contract alonge a different index!]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
yup, *affine* connection means you can translate
grr...no-one told me that!
OH!!! You should think of parallel transport as a function
on the fundamental *groupoid*!
[image of the fundamental group at a point is the holonomy group;
local holonomy group is the one coming from contractible paths;
obviously? only need small paths]
[flat means local holonomy is trivial]
(BTW, not that many holonomy groups: Berger's list)
[btw, I think an "affine manifold" is one with no curvature,
but you're allowed to have torsion;
not sure exactly what the group you're allowed is:
translation ...and dilation?
oops...an affine manifold is a manifold with an affine connection (Cartan);
an affine space is the Kleinian version: it's a symmetric space]
!!!
It's just the 2-form that gives holonomy!
(...and this is the infinitesimal interpretation)
for infinitesimal, easiest is to think of an infinitesimal loop around that point;
more concretely, take a loop that starts at p, goes out by epsilon radially,
goes around a circle, then comes back
(picture obviously);
the radial moves "cancel" in the sense that it's a conjugation
(you're measuring the holonomy around the circle,
identifying the v.b. over the base point of the circle with the v.b. at p
via the path. there, happy?)
NB: also get holonomy over topologically non-trivial loops
It's a 2-form b/c homotopy of paths is 2D (path = 1D, homotopy = 1D)
[EG, angle/d\theta is a flat but not trivial connection on the line bundle over R^2\0:
it's locally trivial but not globally trivial.
holonomy over a null-homotopic curve is integral of curvature over a Seifert surface
(this generalizes Gauss-Bonnet, and is rather like Stokes' theorem);
over a non null-homotopic curve, you can compute it as
. the holonomy over a representative curve in that homology class
. + the integral of curvature over a homology between 'em]
[note that for surfaces, the only metric curvatures are rotation left/right,
hence can be identified with a single number: so(2) = R
*that's* why 2D curvature is zero.
oh, and 1D curvature is trivial, since so(1) = 0
(yes, I'm using a Lie algebra; it's infinitesimal, fool.)
The point is that in 3D, curvature is *not* a number:
even in a single plane, the curvature is in so(3).
You can't just use your 2D intuition and look at local surfaces in that plane
(er, surface to which this is a tangent plane)]
NB: for the discrete models (cone, corkscrew), the curvature (@ a point)
is an element of SO(2);
for continuous cases, it's infinitesimal, so an element of so(2).
cool!
positive curvature = you rotate in the direction of the orientation;
negative curvature = you rotate in the opposite direction
[in other ways positive and negative curvature are not analogous:
positive has O(2); negative has O(1,1)]
Riemann-Cartan manifold
\section{Models}
Some models to illustrate this intepretation.
discrete:
orbifolds, flat/torsion-free except at a point
- cone: curvature
- corkscrew: torsion
"like a screw dislocation in a crystal"
(parking garage ramp)
(covering space)
continuous:
sphere
(don't know continuous torsion; less familiar with torsion)
\section{Historically}
(historically, people saw this:
- in coordinates
- in terms of developments (rolling planes along surfaces)
(which is part of why the term ``affine'' is abused:
thought of tangent space at point as an affine space with a marked point
[which in retrospect tells you that it's a vector space],
and of development as an affine map)
)
\section{Lie theory}
In Lie algebras, everything coincides, so it's confusing.
The covariant derivative is aka the adjoint representation:
$\ad_X := [X,-] = \nabla_X$
\section{Cartan formalism}
I haven't thought deeply about the Cartan formalism.
It is very elegant algebra, and a prior inscrutible.
The below is speculation.
You define an $\End TM$-valued $1$-form $\omega \in \Omega^1(\so_n)$,
namely ``covariant derivative in the $v$ direction'',
then define the curvature form as
$\Omega := d\omega + \omega \wedge \omega$.
The usual definition of curvature is as
$$R(u,v)=\nabla_u\nabla_v - \nabla_v \nabla_u - \nabla_{[u,v]}$$
and I think the correspondence is
\begin{align*}
\omega \wedge \omega &= \nabla_u\nabla_v - \nabla_v \nabla_u\\
d\omega &= \nabla_{[u,v]}\\
\end{align*}
actually, maybe that's backwards:
I want to say:
$\int_\gamma \omega$ is holonomy
$\int_A \Omega$ is holonomy
$\int_A d\omega = \int_{\partial A} \omega$ is holonomy
...but I'm missing the $\omega \wedge \omega$
oh!!!
this holds over an isometrically immersed surface
(surface version of geodesic)
...where holonomy fixes the *plane*
The $\omega \wedge \omega$ is to deal with the fact that you move the plane
oic: $\omega \wedge \omega$ is the induced action on $\Lambda^2 E$
(planes in $E$); for a *vector-valued* 1-form, $\omega \wedge \omega \equiv 0$,
but for an *endomorphism-valued* one, it isn't.
why is $\Omega \wedge \theta = 0$?
$\theta$ is the identity $1$-form;
it takes a vector to itself