Commit fe111988 authored by Jannes Bantje's avatar Jannes Bantje

Merge branch 'master' into characteric-and-genera

# Conflicts:
#	contents/elliptic-genera.tex
#	index_theory.tex
parents 472a4391 462f95cd
Pipeline #60038 canceled with stages
......@@ -89,7 +89,7 @@ We denote by $\Gamma(M,E)$ the space of smooth sections of $E$.
The simplest definition of a differential operators goes as follows:
\begin{definition}
\begin{definition}\label{def:diffop-first-order}
A \Index[differential operator!first-order]{first-order linear differential operator} on $E$ is a complex-linear map
\[
D \colon \Gamma(M,E) \longrightarrow \Gamma(M,E)
......@@ -107,7 +107,7 @@ The simplest definition of a differential operators goes as follows:
Although first-order operators are sufficient for most our purposes, here is a more general definition, which uses multi-index notation, see \cref{sec:multi_index_notation}:
\begin{definition}
\begin{definition}\label{def:diffop-general}
Let $E_i \to M$, $i=0,1$, be smooth vector bundles over $\mathbb{K} \in \set*{\mathbb{R},\mathbb{C}}$.
A map $D \colon \Gamma(M,E_0) \to \Gamma(M,E_1)$ is a \Index[differential operator!order $k$]{differential operator of order $k$}, if $D$ is linear and
\begin{enumerate}[(i)]
......@@ -222,7 +222,7 @@ In other words $\benbrace*{D,g}$ is acting via the following element of $C_c^\in
The computation in the proof of \cref{lem:symbol_welldefined} now shows:
\begin{lemma}
If $D$ is an operator of order $1$, the symbol can be computed as\marginnote{again, depending on notation the $i$-term might not come up}
If $D$ is an operator of order $1$, the symbol can be computed as\marginnote{again, depending on conventions the $i$-term might not come up}
\[
\sigma_1(D)(\mathd g) u = i \,\benbrace*{D,g} u
\]
......@@ -240,6 +240,71 @@ The symbol gives raise to an extremly important class of differential operators:
\todo[inline]{further properties of the symbol from \textcite[Sec.~2.2]{ebert_index_lec}?}
% section differential_operators (end)
\section{Differential Operators: A coordinate-independent Description using Jet Bundles} % (fold)
\label{sec:diffops_coordinate_independent_description}
In \cref{def:diffop-first-order,def:diffop-general} we resorted to using local coordinates to define the notion of a differential operator.
In this section shall be a short excursus, how coordinate-independent descriptions can be obtained (different from the one in \cref{thm:diff_op_coordfree}, which takes the viewpoint of commutative algebra).
The main notion needed for this, is a \emph{jet}.
We consider a smooth fibre bundle $\pi \colon E \to M$ and sections $\sigma, \eta \in \Gamma(E)$.
The idea is to compare the derivatives of such sections up to a certain order $r \in \mathbb{N}$.
Given some point $p \in M$ we locally define
\[
\sigma \sim_p \eta \enspace \coloniff \diff{^\alpha \sigma}{x_\alpha} \bigg|_p = \diff{^\alpha \eta}{x_\alpha} \bigg|_p \text{ for all $\abs*{\alpha}\le r$}
\]
Clearly, this is an equivalence relation and the equivalence classes only depend on germs.
\begin{definition}\label{def:jet-of-section}
The equivalence class of $\sigma$ is called the \Index[jet]{$r$-jet} of $\sigma$ at $p$ and is denoted by $j^r_p \sigma$.
\end{definition}
It is sometimes helpful to think of $j^r_p \sigma$ as the Taylor polynomial of $\sigma$ at $p$ of degree $r$.
\begin{propositiondef}\label{def:jet-manifold}
The set
\[
J^r(E) = \set*{j^r_p \sigma \given p \in M, \sigma \in \Gamma(x)}
\]
is called the \Index[jet manifold]{$r$-th jet manifold} of $\pi \colon E \to M$ and in fact a manifold.
\end{propositiondef}
Jet manifolds come with canonical maps $\pi^r \colon J^r(E) \to M$ and $\pi^{r,0} \colon J^r(E) \to E$.
There are also forgetful maps $J^r(E) \to J^s(E)$ for $s \le r$, which fulfil the relations one would expect.
\begin{remark}
These forgetful maps can be used to take the projective limit
\[
J(E) \coloneqq \Plim J^r(E)
\]
However the category of $\infty$-dimensional manifolds in which one takes this limit is not obvious and there are inequivalent choices one could take here!
The object $J(E)$ is therefore not immediately useful.
\end{remark}
\begin{definition}\label{def:jet-bundle}
The triple $\pi^r \colon J^r(E) \to M$ is a fibre bundle, called the \Index{jet bundle}.
The map $j^r \colon \Gamma(E) \to \Gamma(J^r(E))$, which sends sections $\sigma$ to their $r$-jet $j^r \sigma$, is called \Index{jet prolongation}.
\end{definition}
Now we come back to the original goal of defining differential operators in a coordinate-independent fashion.
\begin{definition}[sidebyside,righthand width=3.5cm]\label{def:diffop-via-jet-bundle}
A linear map $P \colon \Gamma(E) \to \Gamma(F)$ is a \Index[differential operator!order $k$]{differential operator of order $k$}, if it factors through the jet-bundle $J^k(E)$, i.e. there is a vector bundle map $i \colon J^r(E) \to F$ such that the diagram on the right commutes.
\tcblower
\begin{tikzcd}
\Gamma(E) \rar["j^r"] \drar["P"'] & \Gamma(J^r(E)) \dar["i_*"] \\
& \Gamma(F)
\end{tikzcd}
\end{definition}
By the way the jet bundle was set up, it is obvious, that locality as in \cref{def:diffop-general} is fulfilled.
\Cref{def:diffop-general} (ii) also holds, since the value of $P \sigma$ clearly depends only on the derivatives of $\sigma$ of order at most $k$.
\todo[inline]{differential relations and formulation of the $h$-principle}
\todo[inline]{Are there cases, where \cref{def:diffop-via-jet-bundle} is actually useful?}
% section differential_operators_a_coordinate_independent_description (end)
\section{Differential Operators: Hilbert Space Techniques and the Formal Adjoint} % (fold)
\label{sec:hilbert_space_techniques_and_the_formal_adjoint}
From now on we assume, that the manifold $M$ is equipped with a Riemannian metric (or some other smooth measure) and that each of our complex bundles is equipped with a Hermitian bundle metric.
......
......@@ -6,10 +6,10 @@
\includeonly{% comment out as needed
contents/diffop,
% contents/K-homology,
contents/K-homology,
contents/characteristic,
contents/elliptic-genera,
% contents/atiyah_singer,
contents/atiyah_singer,
contents/coarse,
contents/appendix
}
......
......@@ -38,15 +38,54 @@
% -- K für K-Theorie
% ======================================================================================
% \makeatletter
% \newcommand*{\IfBold}{%
% \ifx\f@series\my@test@bx
% \expandafter\@firstoftwo
% \else
% \expandafter\@secondoftwo
% \fi
% }
% \newcommand*{\my@test@bx}{bx}
% \makeatother
%-- IfBold Command
\makeatletter
\newcommand*{\IfBold}{%
\ifx\f@series\my@test@bx
\expandafter\@firstoftwo
\DeclareRobustCommand\IfFontSeriesContextTF[1]{%
\expand@font@defaults
\@font@series@contextfalse
\def\requested@test@context{#1}%
\expandafter\edef\csname ??def@ult\endcsname{\f@family}%
\let\@elt\test@font@series@context
\@meta@family@list
\@elt{??}%
\let\@elt\relax
\if@font@series@context
\expandafter\@firstoftwo
\else
\expandafter\@secondoftwo
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\fi
}
\newcommand*{\my@test@bx}{bx}
\def\test@font@series@context#1{%
\edef\reserved@a{\csname #1def@ult\endcsname}%
\ifx\f@family\reserved@a
\let\@elt\@gobble
\expandafter\ifx
\csname\requested@test@context series@#1\endcsname\f@series
\@font@series@contexttrue
\else
\expandafter\ifx
\csname\requested@test@context def@ult\endcsname\f@series
\@font@series@contexttrue
\fi\fi\fi
}
\newif\if@font@series@context
\newcommand\IfBold{\IfFontSeriesContextTF{bf}}
\makeatother
\newcommand{\Kay}{\IfBold{\ensuremath{\mathbold{K}}}{\ensuremath{K}}}
\newcommand{\Eee}{\IfBold{\ensuremath{\mathbold{E}}}{\ensuremath{E}}}
......
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