\chapter{Fundamental structures of differential geometry}
There is a bewildering array of structures in differential geometry.
For our purposes,
the basic structure on a smooth manifold is the derivative of functions,
and
the basic structure on a Riemannian manifold is the metric.
From these, you get a zoo:
you can do many computations and operations on a smooth manifold,
and many more on a Riemannian manifold.
To organize this, we isolate a few fundamental intermediate structures
(derived from the derivative and the metric)
from which the others follow easily.
These structures are all maps between some combination of functions, vector fields, and covector fields ($1$-forms).
They are defined infinitesmially, but not always pointwise:
they are bundle maps on the jet bundle, but not always on the (co)tangent bundle:
the value at a point depend on the value \emph{and derivatives} of the
function/vector field/form at a point.
[these other structures are called \Def{Jets}
(maps between jet bundles); in this case they are differential operators]
A structure that depends only on the value (and not derivatives) is called
\Def{tensorial} or a \Def{tensor}.
\section{Notation}
Let $C^\infty M$ be the algebra of functions on $M$, aka the regular functions on $M$
(also denoted $\cF M$ or $\cO_M$).
Let $\cT M$ and $\cT^* M$ denote smooth sections of the tangent and cotangent bundles
(also denoted $\Gamma(TM)$ and $\Gamma(T^*M)$ for ``globally sections'');
these are modules over $C^\infty M$, as is $C^\infty M$ itself.
Indeed, they are locally free (as vector bundles are locally trivial),
but not vector spaces (the algebra of scalars is not a field!).
The structures are all maps between tensor products (over $C^\intfy M$
or just over $K$) of these modules; I don't know a proper name for these.
Note that $\cT^* M$ is the \emph{fibrewise} dual of $\cT M$;
I don't know if it's the dual as modules.
You can also write $\Omega^1 M = \cT^* M$ ($1$-forms are covector fields).
\subsection{Coordinates and coefficients}
We can write these structures in coordinates.
For concreteness, write a structure as a map
$$A\from \cT^* M \times \dots \times \cT^* M \times \cT M \times \dots \times \cT M \to K $$
Take a basis for the tangent bundle, meaning a basis as $C^\infty M$-modules;
concretely, $n$ vector fields $e_i$ that are linearly independent at each point
(that are a basis fiberwise); this also yields a basis for the cotangent bundle.
This can always be done locally, by local triviality of vector bundles.
In physics, this is called a \Def{moving frame}.
Now \emph{given a basis}, the value of $A$ over a point $m \in M$ depend
only on the values at $m$ (this is special to the structures we discuss),
so we get a fibrewise map
$$A(m)\from T^*_m \times \dots \times T^*_m \times T_m \times \dots \times T_m \to K$$
Then the coefficients of $A$ are the coefficients in the basis induced by $e_i$;
the coefficients are scalars, elements of $C^\infty M$ (functions of $m$, not constants).
For instance if $A(m)\from T^*_m \times T^*_m \times T_m \to K$,
then $e^i\otimes e^j \otimes e_k$ is a basis for $V^* \otimes V^* \otimes V$,
and we write $a^{ij}_k = A(e^i,e^j,e_k)$.
If $A$ is being interpreted as outputting vectors (or covectors),
you might instead write this as $A(e^j,e_k) = a^{ij}_k e_i$.
\subsection{Examples}
\begin{description}
\item[covariant metric tensor] $g_{ij} = g(e_i,e_j)$
\item[contravariant metric tensor] $g^{ij} = g(e^i,e^j)$
\item[Torsion tensor] (using the below) $T^c_{ab} = \Gamma^c_{ab} - \Gamma^c_{ba}-\gamma^c_{ab}$
\item[Riemann curvature] $R^l_{ijk}$
\end{description}
This also applies for some operations that are not tensorial, for instance:
\begin{description}
\item[Christoffel symbols]
$$\nabla_ie_j=\Gamma_{ij}^ke_k$$
where $\nabla_i e_j$ is the covariant derivative.
Equivalently,
$$\Gamma_{ij}^k = e^k\nabla_ie_j$$
\item[commutator coefficients]
$$[e_i,e_j] = \gamma_{ij}^k e_k$$
where $[e_i,e_j]$ is the Lie bracket.
Equivalently,
$$\gamma_{ij}^k = e^k[e_i,e_j].$$
\end{description}
\subsection{Transform}
how do coefficients transform under change of coordinates?
``tensorial''
*transform*
where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiatio, as in: $v^i_{;j} := \nabla_j v^i$
This is a warning that it's not tensorial
(it's linear over $C^\intfy M$ in $j$, but not in $i$).
Note that the antisymmetrized covariant derivative ∇uv - ∇vu, and the Lie derivative Luv differ by the torsion of the connection, so that if a connection is symmetric, then its antisymmetrization is the Lie derivative.
coefficients and coordinates
given vector fields, get coefficients
(if vector fields are holonomic, life is nicer)
mathy terms:
something-valued 2-form
keep track of what is $C^\intfy M$ linear and what isn't
(by formal math way, or by ; notation)
\section{Smooth manifold}
The basic structure on a smooth manifold is differentiation of functions.
On any smooth manifold, you have:
\begin{itemize}
\item the directional derivative of a function (with respect to a vector field)
$$\cT M \times C^\infty M \to C^\infty M$$
This is very concrete and basic: it's differential calculus in several variables.
More abstractly, a vector field gives an operator on functions
(``differentiate with respect to this vector field''),
indeed exactly the first-order differential operators
(or derivations).
Note that $\cT M \times C^\infty M \neq \cT M$: the derivative isn't linear over
$C^\intfy M$ (in the second variable -- it's a derivation, so there is an error term
involving the derivative),
so we can't just say $V \otimes K = V$.
\item the differential of a function
$$C^\infty M \to T^* M$$
This is exactly the same data as the directional derivative, but in a more elegant form.
\item Lie derivative (bracket of vector fields)
$$\cT M \times \cT M \to \cT M$$
This can be defined many ways; as above, one can identify vector fields with derivations,
and then take the bracket of two derivations, which yields another derivation,
which thus comes from a vector field. Less symmetrically, you can take the derivative of a
vector field $Y$ w/r/t a vector field $X$:
$$\cL_X Y := [X,Y] := X(Y(f)) - Y(X(f))$$
Note that the second equality defines $[X,Y]$ as a derivation;
identifying it with a vector field isn't immediate (you need to know that derivations
come from vector fields).
The Lie derivative is \emph{not} a $(1,2)$-tensor: it depends not just on the values
of the vector fields at the point, but the (first order) derivatives (in a basis).
\end{itemize}
\subsection{More on Lie bracket}
********
it's the non-commutativity
what's less obvious is that this difference is *itself* a derivation;
that's a general fact & representability
Lie derivative measures failure to commute;
equivalently, the obstruction to holonomy of 2 vector fields
(since mixed partials are equal)
Also it's non-commutativity of flows.
Given $n$ independent vector fields $e_i$, you can write the commutator coefficients:
$$\gamma^c_{ab} e_c := [e_a,e_b]$$
which are the obstruction to holonomy.
Because the Lie bracket is not tensorial, these coefficients do not transform
as you would expect from the indices.
\section{Riemannian manifold}
The basic additional structure on a Riemannian manifold is the metric;
this is a $(0,2)$-tensor, and is written in coordinates as $g_{ij}$.
Christoffel symbols
Because the covariant derivative is not tensorial, these coefficients do not transform
as you would expect from the indices.
The subtlest structure is a (Koszul) connection:
$$\cT M \times \cT M \to \cT M$$
The space of connections on E is an affine space for Ω1(End E).
On a Riemannian manifold, you have:
- a metric: (0,2)
vf x vf -> f
- a connection
vf x vf -> vf
- a curvature: (1,3)
vf x vf x vf -> vf (!)
torsion (aka, Cartan torsion tensor)
torsion *is* a (1,2) tensor
$$T(X, Y) := \nabla_X Y - \nabla_Y X - [X,Y]$$
$$T^c_{ab} := \Gamma^c_{ab} - \Gamma^c_{ba}-\gamma^c_{ab}$$
Parallel transport is path-dependent;
torsion measures the infinitesimal path dependence
curvature is the rotation;
torsion is the translation
scalar curvature is the Lagrangian
The Levi-Civita connection is the torsion-free connection
Riemannian curvature
Ricci curvature
Scalar curvature
Trace-free Ricci tensor
constant scalar
constant Ricci (Einstein manifolds)
constant sectional
The Lie derivative and covariant derivative are *not* tensors,
but the *torsion* and *curvature* are.
- metric
- connection
- curvature
a connection (and covariant derivative) is *not* a tensor:
it depends on the 1-jet, not just the value at the point
A metric yields
- the Levi-Civita connection
- the Riemannian curvature
Here's the point:
- curvature is the invariant
Associated with this metric tensor, there is a Levi-Civita connection which is torsionless but has a nonzero curvature form (Riemann tensor) in general.
The standard assumption is to make the spin connection torsionless, but it can equally well be chosen to have a nonzero torsion but zero curvature form. The latter choice leads to the Weitzenböck connection. The zero curvature condition means that there is a global moving frame (the parallel transport of the orientation of the tetrads is path independent), i.e. a global orientation.
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BTW, Weyl tensor = traceless part of Riemann curvature tensor
= Riemann - Ricci
It's conformally invariant, not just isometrically invariant
(intuitively b/c dilation affects trace but not the traceless part)
scalar curvature is just "how volume scales"