Commit fe111988 by 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 \expandafter\@secondoftwo \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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