\chapter{Differential forms}
Differential forms are a key tool
A differential $k$-form on $M$ is a section of $\Lambda^k T^*M$;
it is dual to (sums of) $k$-frames.
This is a very elegant and simple definition,
but opaque.
What do you *do* with them?
Well:
- you pair 'em against frame fields
...and often then integrate
- de Rham:
Concretely, you often think of $k$-forms as dual to $k$-cells:
a $k$-form is something that you integrate over a $k$-cell
(a $k$-cell has a tangent vector field;
dually, you pull back the $k$-form to the $k$-cube and integrate
there, where it's a volume form)
This is the de~Rham complex point of view:
look at the pairing between the de~Rham complex
and the singular chain complex.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Duality of vector fields and $1$-forms}
Vector fields form a Lie algebra (under Lie bracket).
Differential $1$-forms form a Lie coalgebra (under derivative).
[Am I missing any technicalities?]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
...but you can also:
- integrate over a different-dimensional space
EG, you can integrate a $2$-form over a curve,
so long as you have a $2$-frame over the curve.
[careful: don't actually need a $2$-frame;
(don't need non-vanishing, linearly independent, etc.)
just need a section of $\Lambda^2 TM$]
This makes me think: often I think of integrating over a surface
that bounds a curve (e.g., curvature),
but can you just directly integrate over the curve?
- partially integrate
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Given an associative algebra, we get a Poisson algebra
(associative, Lie, play nice together).
A Poisson structure on a manifold is a Poisson algebra structure on
the ring of regular functions.
The ring of regular functions on a manifold is an associative algebra,
but it's commutative (booooring): the trivial Poisson structure.
If it's symplectic, you get a Poisson algebra structure on the
ring of regular functions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Remember:
These are natural:
derivative of a function
F -> T^*M
...from which you also get:
Lie derivative of a function by a vector field
F x TM -> F
Lie bracket of two vector fields
TM x TM -> TM
(it's actually Ext^2 TM -> TM)
Lie cobracket of 1-forms (yielding a 2-form)
T^*M -> T^*M x T^*M
(actually T^*M -> Ext^2 T^*M)
What is *not* natural is a map
F -> TM
Now, a non-degenerate form TM -~> T^*M
(like a pseudo-Riemannian metric or symplectic form)
yields this.
Thus you get F -> TM
for (pseudo-)Riemannian, this is called "gradient vector field";
for symplectic it's called "Hamiltonian vector field".
Ok, *now* you also get
F x F -> TM x T^*M -> F
...take gradient/Hamiltonian v.f. of one, and evaluate on derivative of other.
so (at least in symplectic case), you get a Lie bracket on F,
which is a nice derivation of functions so you get a Poisson algebra!
[For symplectic manifolds, Hamiltonian vector fields are closed under
bracket -- is this also true for gradient vector fields??
OIC why this is weird: it's the *Lagrangian* formulation.
So here's the point:
(Lie) bracket of vector fields is ok and understandable;
(Poisson) bracket of functions is Hamiltonian/Lagrangian mechanics
(for symplectic/pseudo-Riemannian manifolds)
...which is why I'm confused: you really need to understand the
mechanics formalism.
]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Geodesic flow is a special case of Hamiltonian flow!
This is called "the Hamilton-Jacobi approach to the geodesic equation".
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Oh!!!!
The (total space of the) cotangent bundle of a manifold has a symplectic structure.
A pseudo-Riemannian metric gives an iso between cotangent bundle and
tangent bundle.
Thus the (total space of the) tangent bundle of a Riemannian manifold
gets a symplectic structure.
Geodesic flow is the Hamiltonian flow on this;
see elsewhere for details.