Riccardo

::

Little” post=98251Preferenze di BibDesk, sezione Files.
Sulla voce: “Open and Export BibTeX files using encoding” è selezionato “UNICODE UTF-8”?

Sì, ho avuto cura di impostare la codifica Unicode appena ho installato BibDesk!

::

Grazie mille, mi era sfuggita quella discussione. Il problema è ora risolto!
Grazie a tutti per l’aiuto e la collaborazione!

::

Grazie mille!

::

Ok, grazie ancora!

Ciao
Riccardo

::

ivan” post=64990

Ho provato, non cambia nulla. 🙁

Perché togliere i mini-indici? L’idea mi piace… forse entrano in conflitto con la bibliografia? Non mi sembra: per gli altri capitoli non è così…

Le note a margine sono al momento solo dei promemoria, mi servono come “appunti” di revisione. In fase finale non ce ne sarà nemmeno una, ma al momento mi occorrono, le tolgo man mano che risolvo certe questioni. So bene che stonano con lo stile del testo! 🙂

Deve funzionare. Prova a mettere i comandi inclusi nel file “bibliografia” direttamente nel main, come ho fatto io.
Ottengo la bibliografia nell’indice correttamente, semplicemente copiando in sequenza i codici che hai fornito.

Ciao
Ivan

La voce “Bibliografia” nell’indice compare correttamente anche a me, così come il numero di pagina relativo. Il problema però non è questo: il collegamento ipertestuale del numero di pagina che dovrebbe condurre alla pagina contenente la bibliografia punta invece altrove.

::

Il problema è che l’ho scritto, ma non sembra funzionare. Se non carico l’ultimo capitolo va tutto a meraviglia, ma con l’ultimo capitolo ho questo problema.

Scusate, pensavo di aver allegato i files, ma evidentemente non l’ho fatto.

Questo è il file principale:

`\documentclass[a4paper, 11pt, headinclude, footinclude, titlepage, drafting]{scrbook} %manca il frontespizio, da fare!

\input{PackagesThesis.tex}
\input{SettingsThesis.tex}

\begin{document}

\frontmatter

\dominitoc
\tableofcontents

\mainmatter

\part{Prerequisites}

% \input{Chapters/1_Homology_Cohomology.tex}
% \input{Chapters/2_Vector_Bundles.tex}
% \input{Chapters/3_CharClasses.tex}

\part{The Fedosov Index Formula}

\input{Chapters/IndexManifold.tex}

\backmatter

% \appendix
%
\input{Others/BiblioBibTeX.tex}

\end{document}`

e questo è il file BiblioBibTeX:

`\cleardoublepage
\addcontentsline{toc}{chapter}{\tocEntry{\bibname}} % \tocEntry serve per far comparire bibliography nello stesso stile degli altri capitoli (nell'indice)
\nocite{*}
%\printbibliography % with biblatex
\bibliography{Others/BiblioDB}`

I files per caricare i pacchetti e le impostazioni sono i seguenti:

`%%%%%%%%%%%%%%%%%%%
% PACCHETTI
%%%%%%%%%%%%%%%%%%%

\usepackage[applemac]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
% \hyphenation{smo-oth}

\usepackage[english]{varioref} % per i riferimenti incrociati con pagina inclusa
\usepackage{enumerate}

%%%%%%%%%%%%%%%%%
% Matematica
%%%%%%%%%%%%%%%%
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
%\usepackage{mathrsfs} % per il mathscr (script)
\usepackage[all]{xypic} % per i diagrammi commutativi
\usepackage{xypdf} % per migliorare i diagrammi di xypic in output

%%%%%%%%%%%%%%%%%%
% Finezze tipografiche e impostazioni di pagina
%%%%%%%%%%%%%%%%%%
\usepackage{mparhack} % finezze tipografiche
\usepackage{fixltx2e} % finezze tipografiche
\usepackage{relsize} % finezze tipografiche

%\usepackage{graphicx} % per le figure

\usepackage{emptypage} % per avere le pagine bianche senza numeri di pagina né testatine
%\usepackage[babel]{csquotes} % consigliato con biblatex
%\usepackage[style=numeric-comp,hyperref]{biblatex} % per la bibliografia (cfr classicthesis)
%\usepackage{makeidx} % per l'indice analitico

\usepackage[eulerchapternumbers, parts, pdfspacing]{classicthesis}
\usepackage{caption} % altrimenti c'è un errore in arsclassica
\usepackage[tight, english]{minitoc} % per i mini-indici nei capitoli
\usepackage{arsclassica}

%%%%%%%%%%%%%%%%%%
% Altro
%%%%%%%%%%%%%%%%%%
\usepackage{eucal} % per il mathcal + figo
\usepackage{mathrsfs} % per il math script
%usepackage{xspace} % per il comando \xspace
\usepackage{xcolor} % per i colori
%\usepackage{lettrine} % per le iniziali di capitolo grandi
%\usepackage[colorlinks=true, breaklinks=true]{hyperref} % per i links (colorlinks: colorati)
%\usepackage[toc]{glossaries} % per l'indice dei simboli
%\usepackage[lowtilde]{url} % per indirizzi internet, già caricato da hyperref in classicthesis`

`%%%%%%%%%%%%%%%%%%%%%%
% TEOREMI E DEFINIZIONI
%%%%%%%%%%%%%%%%%%%%%%

% Serve amsthm

\newtheoremstyle{classicdef}% Nome
{12pt}% Spazio che precede l’enunciato
{12pt}% Spazio che segue l’enunciato
{}% Stile del font dell’enunciato
{}% Rientro (se vuoto, non c’è rientro,
% \parindent = rientro dei capoversi)
{\scshape}% Stile del font dell’intestazione
{.}% Punteggiatura che segue l’intestazione
{.5em}% Spazio che segue l’intestazione:
% ” ” = normale spazio inter-parola;
% \newline = a capo
{}% Specifica l’intestazione dell’enunciato
% (normalmente viene lasciata vuota)

\newtheoremstyle{classicthm}% Nome
{12pt}% Spazio che precede l’enunciato
{12pt}% Spazio che segue l’enunciato
{\itshape}% Stile del font dell’enunciato
{}% Rientro (se vuoto, non c’è rientro,
% \parindent = rientro dei capoversi)
{\scshape}% Stile del font dell’intestazione
{.}% Punteggiatura che segue l’intestazione
{.5em}% Spazio che segue l’intestazione:
% ” ” = normale spazio inter-parola;
% \newline = a capo
{}% Specifica l’intestazione dell’enunciato
% (normalmente viene lasciata vuota)

\theoremstyle{classicthm}
\newtheorem{thm}{Theorem}[chapter]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}

\theoremstyle{classicdef}
\newtheorem{defn}[thm]{Definition}
\newtheorem{ex}[thm]{Example}

\theoremstyle{remark}
\newtheorem*{rmk}{Remark}
\newtheorem*{note}{Notation}

\newcommand{\dfn}[1]{\emph{#1}} % definizioni nel corpo del testo
\newcommand{\titlethm}[1]{#1} % titoletto dei teoremi

%%%%%%%%%%%%%%%%%%%%%
% COMANDI MATEMATICI
%%%%%%%%%%%%%%%%%%%%%%

% Serve amsmath

\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\hat}[1]{\widehat{#1}}
\renewcommand{\tilde}[1]{\widetilde{#1}}

\newcommand{\A}{\mathcal{A}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\GL}{\mathrm{GL}}
\newcommand{\Hdr}{H_{\textup{dR}}}
\newcommand{\Id}{\mathrm{Id}}
\newcommand{\K}{\mathcal{K}}
\newcommand{\M}{\mathrm{M}}
\newcommand{\N}{\mathbb{N}}
\renewcommand{\P}{\mathcal{P}}
\newcommand{\R}{\mathbb{R}}
\renewcommand{\S}{\mathbb{S}}
%\newcommand{\V}{\mathcal{V}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\eps}{\varepsilon}
%\renewcommand{\phi}{\varphi}
%\renewcommand{\empty}{\varnothing}

\renewcommand{\d}{\mathrm{d}}
\renewcommand{\=}{\mathrel{\mathop:}=}
\newcommand{\eq}{=\mathrel{\mathop:}}
\newcommand{\incl}{\hookrightarrow}
\newcommand{\smoothim}[2]{\mathscr{C}^\infty(#1, #2)}
\newcommand{\smooth}[1]{\mathscr{C}^\infty(#1)}
\newcommand{\contim}[2]{\mathscr{C}(#1, #2)}
\newcommand{\cont}[1]{\mathscr{C}(#1)}
\newcommand{\VS}{\mathsf{VS}}
\newcommand{\VB}{\mathsf{VB}}
\newcommand{\Vect}{\mathrm{Vect}}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Iso}{\mathrm{Iso}}

\renewcommand{\vref}[1]{\ref{#1} \vpageref{#1}} % per non avere “on page” in blu nei collegamenti
\newcommand{\veqref}[1]{\eqref{#1} \vpageref{#1}} % per avere un richiamo rapido di formula con pagina

\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\ch}{Ch}
\DeclareMathOperator{\ind}{ind}
%\DeclareMathOperator{\Hom}{Hom}
%\DeclareMathOperator{\End}{End}
%\DeclareMathOperator{\Vect}{\mathsf{Vect}}
%\DeclareMathOperator{\VB}{\mathsf{VB}}
%\let\d\relax
%\DeclareMathOperator{\d}{d}

%%%%%%%%%%%%%%%%%%%%%%%%%
% Mini-indici
%%%%%%%%%%%%%%%%%%%%%%%%%

% Serve minitoc

\def\ptctitle{Contents} % per scrivere “Contents” non in grassetto nei miniindici
\def\mtctitle{Contents}
\def\stctitle{Contents}
\setlength{\mtcindent}{0pt}
\renewcommand{\mtifont}{\normalsize\scshape\lsstyle}

%\renewcommand{\labelenumi}{(\roman{enumi})} % si può sostituire con il pacchetto enumerate

%%%%%%%%%%%%%%%%%%%%%%%
% Bibliografia
%%%%%%%%%%%%%%%%%%%%%%%
%\addbibresource{Others/Bibliography.bib} % include Bibliography.bib CON BIBLATEX!!!
% \renewcommand{\newunitpunct}{, }
\bibliographystyle{amsplain} % stile bibliografico con BibTeX

%%%%%%%%%%%%%%%%%%%%%
% Geometria della pagina
%%%%%%%%%%%%%%%%%%%%%
\areaset[13mm]{\dimexpr 312pt+1.5cm\relax}{699pt} % guadagna 1cm, prendendo lo spazio delle note a margine
\pagestyle{scrheadings} % per avere i numeri di pagina nelle testatine

%%%%%%%%%%%%%%%%%%%%%
% Glossario
%%%%%%%%%%%%%%%%%%%%%
%\makeglossaries`

In ultimo vi scrivo il codice sorgente per l’ultimo capitolo, anche se dubito che sia utile…

`\chapter{Index of an elliptic system on a compact manifold}
\minitoc
\mtcskip

\thispagestyle{empty}

% ————————————————
\section{Setting and Background}
% ————————————————

We assume throughout this chapter that every cited object is smooth, except when differently specified.

We consider a compact manifold $M$ and a vector bundle over it, identified with a family $P \in \mathrm{M}^{n \times n}\big(\smooth{M}\big)$ of projectors; the description is through local
frames and coframes as in Section \vref{framecoframe}. The vector bundle is endowed here with its \textcolor{red}{standard Hermitian metric}%
\marginpar{Define Hermitian metric on vector bundles in the appropriate Chapter}
and we shall deal with orthogonal projection operators. In this case, in fact, i.e. when the matrices $P(x)$ are Hermitian, for a given frame $e_\alpha(x)$ defined in a neighbourhood
$U_\alpha$, the dual coframe is uniquely determined by the formula
\[
\eta_\alpha(x) = \big(e_\alpha(x)^\dag e_\alpha(x)\big)^{-1}e_\alpha(x)^\dag,
\]
where the symbol $\dag$ denotes Hermitian conjugation. Moreover, this choice will not affect the generality of the subject, by virtue of the following lemma. We omit henceforth the dependence
on the point $x$ (with the understanding that each object is actually a matrix) and also on the neighbourhood $U_\alpha$.

\begin{lemma}
Any projector $P$ is homotopic to an orthogonal projector $P'$.
\end{lemma}

\begin{proof}
Given a frame $e$, let $P = e \eta$; we construct an orthogonal projector $P'$ by putting
\[
P' \= e(e^\dag e)^{-1}e^\dag.
\]
We have then that $PP' = P'$ and $P'P = P$:
\begin{gather*}
PP' = e\eta e(e^\dag e)^{-1}e^\dag = e(e^\dag e)^{-1}e^\dag = P', \\
P'P = e(e^\dag e)^{-1}e^\dag e \eta = e \eta = P.
\end{gather*}
Thus the convex combination $P(t) \= tP' – (1 – t)P$ is a projector for all $t \in [0,1]$, and this is the required (straight-line) homotopy between $P$ and $P'$.
\end{proof}

When taking into consideration the curvature matrix associated with a connection on the vector bundle $P$, we shall denote the multiplication of two matrices whose elements are differential
forms on $M$ just as the usual matrix multiplication, i.e. without an interposed sign. By the way, if $A \= (a_i^j)$ is a matrix of $p$-forms and $B \= (b_i^j)$ is a matrix of $q$-forms, the generic
element of their product is given by $c_i^j \= a_i^k \wedge b_k^j$. We shall also denote by $\d A$ the matrix with entries $\d a_i^j$.

The following identities follow directly from the properties of exterior derivative, trace and Hermitian conjugation:
\begin{gather}
\d(AB) = \d A\, B + (-1)^p A\,\d B, \\
\tr(AB) = (-1)^{pq}\tr(BA), \label{tr}\\
(AB)^\dag = (-1)^{pq}(BA)^\dag.
\end{gather}

The Levi-Civita connection on $T^*M$ is obtained%
\marginpar{Show this? On $TM$ or $T^*M$? See Park, where to find the correlation algebraic–differential?}
by embedding the manifold $M$ in $\C^N$ for some appropriate natural number $N$ by Whitney theorem, then applying the ordinary
directional derivative (which is a connection in $\C^N$) and finally projecting the result onto the tangent bundle, seen as a subbundle of $\Theta^N(M)$.

\begin{defn}\label{CovDiff}
A \dfn{vector form}%
\footnote{We call a \dfn{vector form} a (column) vector of differential forms.}
$\vartheta$ is said to be \dfn{invariant under $P$} if
\[
P\vartheta = \vartheta.
\]
Similarly, a \dfn{matrix form}%
\footnote{We call a \dfn{matrix form} a (square) matrix of differential forms.}
$A$ is said to be \dfn{invariant under $P$} if
\[
PA = AP = A.
\]
\end{defn}

With the notation adopted thus far, we give the following definition.

\begin{defn}
The \dfn{covariant differential} of a vector form $\vartheta$ invariant under $P$ is %
\marginpar{Show this? Compare algebraic definition (Park, p.~188) with geometric definition (Ch. 2)}
\[
\nabla \vartheta \= P\, \d\vartheta,
\]
and that of a matrix form $A$ invariant under $P$
\[
\nabla A \= P\, \d A\, P.
\]
\end{defn}

The curvature of this Levi-Civita connection is then given by
\begin{align*}
\nabla^2 \vartheta & = P\, \d (P\, \d\vartheta) = P\, \d P\, \d(P\vartheta) \\
& = P\, \d P\, \d P\, \vartheta + P\, \d P\, P \d\vartheta.
\end{align*}
In order to simplify this expression, we make use of the following lemma.

\begin{lemma}\label{PdPP}
We have the equality
\begin{equation} \label{PdPP:eq}
P\, \d P\, P = 0.
\end{equation}
Furthermore, the matrices $P$ and $\d P\, \d P$ commute.
\end{lemma}

\begin{proof}
By differentiating the idempotence relation $P^2 = P$, we obtain:
\[
\d P\, P + P\, \d P = \d P
\]
and multiplying by $P$ on the left (the same result is anyhow achieved by multiplication on the right) we get
\[
P\, \d P\, P + P^2\, \d P = P\, \d P,
\]
which simplifies to \eqref{PdPP}. The commutation property follows immediately from differentiation of \eqref{PdPP}:
\[
\d P\, \d P\, P – P\, \d P\, \d P = 0. \qedhere
\]

\end{proof}
The curvature now reads
\[
\nabla^2 \vartheta = P\, \d P\, \d P\, \vartheta \eq \Omega\vartheta,
\]
where $\Omega$ denotes the curvature matrix of $\nabla$. From the previous lemma we deduce that any integer power $\Omega^m$ is invariant under $P$: indeed, because of the
commutativity of $P$ and $\d P\, \d P$, $\Omega$ can be represented in the forms
\[
\Omega \= P\, \d P\, \d P = P^2 \d P\, \d P = P\, \d P\, \d P\, P = \d P\, \d P\, P,
\]
whence $P\Omega = \Omega P = \Omega$.

\begin{lemma}
The covariant differential of $\Omega^m$ vanishes for all $m \in \N$.
\end{lemma}

\begin{proof} \marginpar{See Park}
We have, for $m = 1$:
\[
\d\Omega = \d(P\, \d P\, \d P\, P) = \d P\, \Omega + \Omega\, \d P,
\]
whence, by Lemma \ref{PdPP} and Definition \ref{CovDiff},
\[
\nabla\Omega = P\, \d P\, \Omega P + P \Omega\, \d P\, P = P\, \d P\, P \Omega + \Omega P\, \d P\, P = 0.
\]
Note that it is sufficient to show that $\nabla\Omega = 0$ (the second Bianchi identity \veqref{Bianchi}), because of the invariance of $\Omega$ under $P$:
\begin{align*}
\nabla\Omega^2 & \= P\, \d\Omega^2\, P \\
& = P(\d\Omega\, \Omega + \Omega\, \d\Omega)P \\
& = P(\d P\, \Omega + \Omega\, \d P)\Omega P + P\Omega(\d P\, \Omega + \Omega\, \d P)P \\
& = P\, \d P\, \Omega^2 P + P\Omega\, \d P\, \Omega P + P\Omega\, \d P\, \Omega P + P \Omega^2 \, \d P\, P \\
& = P\, \d P\, P \Omega^2 + 2 \Omega P\, \d P\, P \Omega + \Omega^2 P\, \d P\, P \\
& = 0
\end{align*}
and by induction one achieves the result.
\end{proof}

\begin{lemma}\label{lemma:dnabla}
Let $A$ be a matrix form invariant under $P$. Then
\[
\d (\tr A) = \tr (\nabla A).
\]
\end{lemma}

\begin{proof} \marginpar{See Park}
By virtue of Lemma \ref{PdPP} again and property \veqref{tr} we have
\begin{align*}
\d (\tr A) & = \d[\tr(PAP)] = \tr[\d(PAP)] \\
& = \tr[\d(PA)P + PA\, \d P] \\
& = \tr(\d P\, AP + P\, \d A\, P + PA\, \d P) \\
& = \tr(\nabla A) + \tr(P\, \d P\, A + A\, \d P\, P) \\
& = \tr(\nabla A) + \tr(P\, \d P\, PA + AP\, \d P\, P) \\
& = \tr(\nabla A). \qedhere
\end{align*}
\end{proof}

We now introduce the $2k$-forms%
% \footnote{The reason for which we divide $\Omega$ by the factor $2\pi i$ is to have integer coefficients in the cohomology classes.}
\[
\varphi_k \= \tr \left( – \frac{\Omega}{2\pi i} \right)^k.
\]
A direct application of the previous two lemmas shows that each $\varphi_k$ is closed, and therefore they define some cohomology classes on $M$. The following result shows that these
last ones do not depend on the vector bundle $P$, but only on its homotopy class.

\begin{lemma}
If $P_1$ and $P_2$ are equivalent vector bundles, then the corresponding forms $\varphi_k^{(1)}$ and $\varphi_k^{(2)}$ are cohomologous.
\end{lemma}

\begin{proof}
Let $P \= P(x, t)$ be a homotopy between $P_1$ and $P_2$. If $\Omega \= \Omega(t)$ denotes the curvature of $P(x, t)$ then we have, by the Fundamental Theorem of Calculus:%
\footnote{Actually, $\Omega$ depends also on $x$, but we already pointed out that $\nabla\Omega^k = 0$.}
\[
\tr\Omega_2^k – \tr\Omega_1^k = \int_0^1 \frac{d}{dt} \tr\Omega^k(t)\, dt.
\]
The aim is to show that the integral on the right-hand side is zero, or, more precisely, that the integrand is an exact matrix form. We have
\[
\tr \frac{d}{dt} \Omega^k(t) = k \tr(\Omega^{k – 1}\dot{\Omega}),
\]
where the dot denotes the differentiation with respect to the parameter $t$. Besides,
\begin{align*}
\dot{\Omega} & = \dot{P}\, \d P\, \d P\,P + P(\dot{\d P}\, \d P + \d P\,\dot{\d P})P + P\, \d P\, \d P\,\dot{P} \\
& = \dot{P}\Omega + P\, \d(\dot{P}\, \d P – \d P\,\dot{P})\,P + \Omega\dot{P}.
\end{align*}
Substituting in the above equation, the first and last term vanish, because $P\dot{P}P = 0$ (by Lemma \ref{PdPP}, with differentiation with respect to $t$ in place of exterior differentiation),
yielding
\[
\tr \frac{d}{dt} \Omega^k(t) = k \tr\{\Omega^{k – 1}P\, \d(\dot{P}\, \d P – \d P\,\dot{P})\,P\}.
\]
Now observe that $P$ commutes with $\dot{P}\, \d P$ and with $\d P\, \dot{P}$: by differentiating $P\, \d P\,P = 0$ with respect to $t$ we obtain%
\marginpar{I can't show this commutativity! By the way, what is the purpose of showing this?}
\[
\dot{P}\, \d P\, P + P\, \dot{\d P}\, P + P\, \d P\, \dot{P} = 0
\]
and applying exterior differentiation to $P\dot{P}P = 0$ gives
\[
\d P\, \dot{P}P + P\, \d\dot{P}\, P + P\dot{P}\, \d P = 0.
\]
Subtracting the two equations and rearranging terms hands: \marginpar{This shows only that $P$ commutes with the difference $\dot{P}\, \d P – \d P\,\dot{P}$}
\[
P(\dot{P}\, \d P – \d P\, \dot{P}) = (\dot{P}\, \d P\, – \d P\, \dot{P})P,
\]
whence it follows that \marginpar{Why should $\dot{P}\, \d P – \d P\, \dot{P} = P(\dot{P}\, \d P – \d P\,\dot{P})P$?}
\[
P\, \d(\dot{P}\, \d P – \d P\,\dot{P})\,P = P\, \d[P(\dot{P}\, \d P – \d P\,\dot{P})P]\,P = \nabla[P(\dot{P}\, \d P – \d P\,\dot{P})P].
\]
Therefore, recalling that $\nabla\Omega = 0$, that $\Omega$ is invariant under $P$ and using Lemma \ref{lemma:dnabla}:
\begin{align*}
\frac{d}{dt}\tr\Omega^k & = k \tr \{\nabla[P(\dot{P}\, \d P – \d P\,\dot{P})P]\Omega^{k – 1}\} \\
& = k \tr \{\nabla[P(\dot{P}\, \d P – \d P\,\dot{P})P\Omega^{k – 1}]\} \\
& = k\, \d\{ \tr [P(\dot{P}\, \d P – \d P\,\dot{P})\Omega^{k – 1}]\},
\end{align*}
which shows the exactness of the integrand.
\end{proof}

From this lemma it follows that the forms $\varphi_k$ define actually \emph{real} cohomology classes on $M$, because in each class of equivalent bundles there exists an orthogonal projector,
which makes the matrix $\Omega$ Hermitian.

The matrix forms $\varphi_k$ can also be expressed via the local curvature matrices $\Omega_j$, each of which is defined in a neighbourhood $U_j$ of a finite open cover $\{U_j\}$ (existing
because of compactness) of $M$. Let $e_j$ and $\eta_j$ be a frame and a coframe in $U_j$, so that $P_j = e_j\eta_j$; then
\begin{align*}
\Omega & \= P\, \d P\, \d P = e_j\eta_j\, \d(e_j\eta_j)\, \d(e_j\eta_j) \\
& = e_j\eta_j(\d e_j\, \eta_j + e_j\, \d\eta_j)(\d e_j\, \eta_j + e_j\, \d\eta_j) \\
& = e_j\eta_j(\d e_j\, \eta_j\, \d e_j\, \eta_j + \d e_j\, \eta_j e_j\, \d\eta_j + e_j\, \d\eta_j\, \d e_j\, \eta_j + e_j\, \d\eta_j\, e_j\, \d\eta_j) \\
& = e_j[(\eta_j\, \d e_j)^2\eta_j + \eta_j\, \d e_j\, \d\eta_j + \d\eta_j\, \d e_j\, \eta_j + \d\eta_j\, e_j\, \d\eta_j] \\
& = e_j[(\eta_j\, \d e_j)^2 + \d\eta_j\, \d e_j]\eta_j + e_j[\d(\eta_j e_j)\d\eta_j] \\
& = e_j[(\eta_j\, \d e_j)^2 + \d\eta_j\, \d e_j]\eta_j \\
& \eq e_j\Omega_j\eta_j,
\end{align*}
where
\[
\Omega_j \= (\eta_j\, \d e_j)^2 + \d\eta_j\, \d e_j = (\eta_j\, \d e_j)^2 + \d(\eta_j\, \d e_j)
\]
is precisely the local curvature matrix in the given frame and coframe in $U_j$. Using property \eqref{tr} of the trace \vpageref{tr}, we obtain the following local expression for the forms
$\varphi_k$:
\begin{equation}\label{phik:local}
\varphi_k = \tr\left( – \frac{\Omega_j}{2\pi i} \right)^k = \left( – \frac{1}{2\pi i} \right)^k \tr[(\eta_j\, \d e_j)^2 + \d(\eta_j\, \d e_j)]^k.
\end{equation}

% —————————————————
\section{The Formula for the Index}
% —————————————————

Let $M$ be an $m$-dimensional Riemannian manifold, let $T^*M$ be its cotangent bundle and denote by $S^*M$ the sphere bundle in $T^*M$. A point in the cotangent bundle can thus
be described by a pair $(s, t)$, where $s \in S^*M$ and $t$ is a non-negative real number. In other words, we split the description of a (co)vector in each fiber into its direction and length. All
points of the kind $(s, 0)$ form the zero section of $T^*M$; for each of these we observe that $s$ is not uniquely defined. It is also convenient to consider the so-called \dfn{infinite section}, that
is the set of all “infinitely distant points'' $(s, +\infty)$, which belong to the Alexandrov compactification \marginpar{Talk about Alexandrov compactification in the appendix?} $\overline{T^*M}$
of the cotangent bundle.

Consider a cover of $\overline{T^*M}$ made up of two open sets only, $U_0$ and $U_\infty$; the former including all “finitely distant points'' (i.e. $U_0 \= T^*M$) and the latter consisting of
all points except for those forming the zero section (i.e. $U_\infty \= \overline{T^*M} \setminus \{(s, t) \in T^*M : t = 0\}$).

We now construct a vector bundle of rank $n \geq m$ over $\overline{T^*M}$, by means of frames and coframes on these two neighbourhoods, in such a way that the transition function on
$S^*M$ coincides with a given non-degenerate matrix-valued function $\tau \colon S^*M \to \GL_n(\C)$.

Let $f_1$ and $f_2$ be smooth real functions of the non-negative real variable $t$, such that
\begin{itemize}
\item $f_1^2 + f_2^2 = 1$;
\item $f_1(t) \= 0$ for all $t \in \big(0, \frac{1}{2}\big)$;
\item $f_1(t) \= 1$ for all $t > 1$
\end{itemize}
(we do not specify their values elsewhere). Define then on $U_0$ the matrices $e_0(s, t) \in \mathrm{M}^{2n \times n}(\C)$ and $\eta_0(s, t) \in \mathrm{M}^{n \times 2n}(\C)$ by setting
\[
e_0(s, t) \= \begin{pmatrix}
f_2(t)\, \Id_{\C^n} \\
f_1(t) \tau(s)
\end{pmatrix}, \qquad
\eta_0(s, t) \= \Big( f_2(t)\, \Id_{\C^n}, f_1(t) \big(\tau(s)\big)^{-1} \Big).
\]
Similarly, on $U_\infty$ we put
\[
e_\infty(s, t) \= \begin{pmatrix}
f_2(t) \big(\tau(s)\big)^{-1} \\
f_1(t)\, \Id_{\C^n}
\end{pmatrix}, \qquad
\eta_\infty(s, t) \= \big( f_2(t) \tau(s), f_1(t)\, \Id_{\C^n} \big).
\]
It is immediate to check that $\eta_0 e_0 = \eta_\infty e_\infty = \Id_{\C^n}$. In the intersection $U_0 \cap U_\infty$, which corresponds to the positive real semi-axis $0 < t < +\infty$, we have the relations \[ e_0(s, t) = e_\infty(s, t) \tau(s), \quad \eta_0(s, t) = \big( \tau(s) \big)^{-1} \eta_\infty(s, t), \] which make $e_0$, $\eta_0$, $e_\infty$ and $\eta_\infty$ frames and coframes of a vector bundle over $U_0$ and $U_\infty$ respectively, with transition function $\tau$ (see \veqref{transf}). The family $P$ of projectors associated with the vector bundle has therefore the form \begin{align}\label{vbundle:constrP} P(s, t) & \= e_0(s, t)\eta_0(s, t) = e_\infty(s, t)\eta_\infty(s, t) \\ & = \begin{pmatrix} f_2^2(t)\, \Id_{\C^n} & f_1(t) f_2(t) \big(\tau(s)\big)^{-1} \\ f_1(t) f_2(t) \tau(s) & f_1^2(t)\, \Id_{\C^n} \end{pmatrix}. \end{align} In order to find the expression for the forms $\varphi_k$, we use formula \veqref{phik:local}. We then compute, removing the dependence on $s$ and $t$ but still bearing it in mind, \[ \d e_0 = \begin{pmatrix} \d f_2\, \Id_{\C^n} \\ \d f_1\, \tau + f_1\, \d\tau \end{pmatrix}, \] whence \[ \eta_0\, \d e_0 = f_2\, \d f_2\, \Id_{\C^n} + f_1\, \d f_1\, \Id_{\C^n} + f_1^2\, \tau^{-1} \d\tau = f_1^2\, \tau^{-1} \d\tau, \] because $f_1\, \d f_1 + f_2\, \d f_2 = 0$, as obtained by differentiating the first property of the functions $f_1$ and $f_2$. Therefore, \begin{align*} (\eta_0\, \d e_0)^2 + \d(\eta_0\, \d e_0) & = (f_1^2\, \tau^{-1} \d\tau)^2 + \d(f_1^2\, \tau^{-1} \d\tau) \\ & = f_1^4(\tau^{-1} \d\tau)^2 + \d f_1^2\, \tau^{-1} \d\tau + f_1^2\, \d\tau^{-1} \d\tau \\ & = f_1^4(\tau^{-1} \d\tau)^2 + \d f_1^2\, \tau^{-1} \d\tau - f_1^2 (\tau^{-1}\d\tau)^2 \\ & = f_1^2(f_1^2 - 1) (\tau^{-1}\d\tau)^2 + \d f_1^2\, \tau^{-1} \d\tau \\ & = - f_1^2f_2^2 (\tau^{-1}\d\tau)^2 + \d f_1^2\, \tau^{-1} \d\tau, \end{align*} where, in the third equality, we used the fact that $\d\tau^{-1} = - \tau^{-1}\d\tau\, \tau^{-1}$, as is readily seen by differentiating the identity $\tau\tau^{-1} = \Id_{\C^n}$. Thus, observing that all powers of $\d f_1^2$ higher than $1$ are annihilated and using the binomial formula, we have \begin{align*} \varphi_k & = \left( -\frac{1}{2\pi i} \right)^k \tr[ - f_1^2 f_2^2 (\tau^{-1}\d\tau)^2 + \d f_1^2\, \tau^{-1} \d\tau]^k \\ & = \left( \frac{1}{2\pi i} \right)^k [(f_1f_2)^{2k} \tr(\tau^{-1}\d\tau)^{2k} - k (f_1f_2)^{2k - 2} \d f_1^2 \tr(\tau^{-1} \d\tau)^{2k - 1}]. \end{align*} This expression can be further simplified if we note that, by associativity of matrix product and property \veqref{tr}, \begin{align*} \tr(\tau^{-1}\d\tau)^{2k} & = \tr[(\tau^{-1}\d\tau)^{2k - 1} (\tau^{-1}\d\tau)] \\ & = \tr[(\tau^{-1}\d\tau)(\tau^{-1}\d\tau)^{2k - 1}] \\ & = - \tr[(\tau^{-1}\d\tau)^{2k - 1} (\tau^{-1}\d\tau)], \end{align*} which in its turn implies that this term must vanish. Moreover, the form $\tr(\tau^{-1} \d\tau)^{2k - 1}$ is closed, because \[ \d[\tr(\tau^{-1} \d\tau)^{2k - 1}] = \tr[\d(\tau^{-1} \d\tau)^{2k - 1}] = - \tr(\tau^{-1} \d\tau)^{2k} = 0, \] the second equality easily following by induction and the expression for $\d\tau^{-1}$ exploited before. With all this information, the forms $\varphi_k$ can now be written as \[ \varphi_k = - \frac{k}{(2\pi i)^k}\, \d \left(\int_0^t {\big(f_1(s)f_2(s)\big)^{2k - 2}\, \d f_1^2(s)} \cdot \tr(\tau^{-1} \d\tau)^{2k - 1} \right). \] For $t > 1$ we have
\[
\int_0^t {\big(f_1(s)f_2(s)\big)^{2k – 2}\, \d f_1^2(s)} = \int_0^1 {[\xi(1 – \xi)]^{k – 1}\, \d \xi} = B(k, k) = \frac{[(k – 1)!]^2}{(2k – 1)!},
\]
where $B$ is the beta function. Then, setting
\[
h_k(t) \= \frac{(2k – 1)!}{[(k – 1)!]^2}\int_0^t {\big(f_1(s)f_2(s)\big)^{2k – 2}\, \d f_1^2(s)},
\]
the forms $\varphi_k$ now read
\[
\varphi_k = – \frac{1}{(2\pi i)^k}\frac{k!(k – 1)!}{(2k – 1)!}\, \d[h_k\tr(\tau^{-1} \d\tau)^{2k – 1}]
\]
and are compactly supported in $(0,1]$, because $h_k(t) = 1$ for all $t > 1$.

\begin{rmk}
We could replace \marginpar{Show this?} $h_k$ by a smooth function $h$, which vanishes in a neighbourhood of the origin and is equal to $1$ for $t > 1$, without affecting the
cohomology class of $\varphi_k$.
\end{rmk}

We then introduce the closed forms on $S^*M$
\[
\psi_k \= \frac{1}{(2\pi i)^k} \frac{(k – 1)!}{(2k – 1)!} \tr(\tau^{-1}\d\tau)^{2k – 1},
\]
so that the Chern character of the vector bundle $P$ has the expression
\[
\ch P = n – \d\! \left( h(t) \sum_{k = 1}^m \psi_k \right).
\]

\begin{rmk}
Since the family $P$ of projectors is constant in a neighbourhood of the zero and infinite sections of $\overline{T^*M}$, we may assume that $P$ is defined on the suspension
\marginpar{Talk about suspension in the appendix? Collapsing construction?} $S(S^*M)$ of $S^*M$. We nowhere used the fact that $S^*M$ is a sphere bundle and therefore everything
we said keeps being true if $S^*M$ is replaced by any (compact?) manifold and $\overline{T^*M}$ by the suspension $S(S^*M)$.

In particular, a given map $\theta \colon \S^{2m – 1} \to \GL_{2m – 1}(\C)$ \marginpar{Is $\GL_{2m – 1}(\C)$ correct?} defines a vector bundle $P$ (as in \veqref{vbundle:constrP}) over the
sphere $\S^{2m}$, which is indeed the suspension of $\S^{2m – 1}$. From the Bott periodicity theorem \marginpar{State this theorem? Where? K-Theory or appendix?} it follows that the
only stable homotopy invariant of $\theta$ is the value of $\ch P$ on $\S^{2m}$. Integrating $\d(h\psi_m)$ over $\S^{2m}$ and applying the Stokes theorem we find that in this case
\[
\ch P = \pm \frac{1}{(2\pi i)^m} \frac{(m – 1)!}{(2m – 1)!} \int_{\S^{2m – 1}} \tr(\theta^{-1}\d\theta)^{2m – 1}.
\]
\end{rmk}

We now consider an elliptic differential operator $L \colon \smoothim{M}{\C^n} \to \smoothim{M}{\C^n}$. Its symbol $\sigma$, considered on $S^*M$, is a non-degenerate matrix-valued
function and therefore it defines a vector bundle of rank $n$ represented through a family $P$ of projectors on $T^*M$, which is constant outside some compact subset of $T^*M$.

The differential form $\ch \sigma$, which has compact support on $T^*M$, is defined as
\[
\ch\sigma \= \ch P – n,
\]
where $n$ is the rank of $P$, as before.

The Atiyah-Singer theorem \marginpar{Write about it? Here, before, in the appendix, as an aside in another chapter,\dots?} gives now an expression for the index of $L$, relating it to the value
of the $2m$-dimensional component of the product $(-1)^m\ch\sigma \cdot \mathscr{T}(M)$ on the fundamental cycle \marginpar{What is the fundamental cycle?} on $T^*M$:
\[
\ind L = (-1)^m \int_{T^*M} \big(\ch\sigma \cdot \mathscr{T}(M) \big)_{2m}.
\]
The integrand is recovered through the expression of $\ch P$ and the Todd class \textcolor{red}{(insert reference to the Todd class here)}:
\[
\big(\ch\sigma \cdot \mathscr{T}(M) \big)_{2m} = – \d\left\{ h(t) \left( \psi_m + \sum_{k = 1}^{\lfloor m/4 \rfloor} \psi_{m – 2k} \mathscr{T}_k \right) \right\},
\]
We observe that this form vanishes for $t > 1$; therefore, we can integrate it over the bundle $B^*M$ of unit balls of $T^*M$, i.e. over the set of all $t \leq 1$, and this yields
\[
\ind L = (-1)^{m + 1} \int_{B^*M} \d\left\{ h(t) \left( \psi_m + \sum_{k = 1}^{\lfloor m/4 \rfloor} \psi_{m – 2k} \mathscr{T}_k \right) \right\}.
\]
Applying the Stokes theorem we finally obtain
\[
\ind L = (-1)^{m + 1} \int_{S^*M} \left( \psi_m + \sum_{k = 1}^{\lfloor m/4 \rfloor} \psi_{m – 2k} \mathscr{T}_k \right).
\]
In the case where all the $\mathscr{T}_k$ vanish \marginpar{Insert conditions for the Todd class to vanish, explain intuitively what it means.} we have the simpler form
\[
\ind L = \frac{(-1)^{m + 1}}{(2\pi i)^m} \frac{(m – 1)!}{(2m – 1)!} \int_{S^*M} \tr(\sigma^{-1}\d\sigma)^{2m – 1}.
\]`