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Update on Overleaf.

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juso.koc 2025-03-21 12:41:01 +00:00 committed by node
parent 62f8a1c05f
commit 28cb8dd720
2 changed files with 4 additions and 4 deletions

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\Delta f \coloneq \divergence \nabla f. \Delta f \coloneq \divergence \nabla f.
\end{align*} \end{align*}
\end{definition} \end{definition}
\item The spectrum of this operator is a subset of $[0, \infty)$. We call it the \emph{spectrum} of the manifold. \item The spectrum of this operator is a subset of $[0, \infty)$. We call it the \textcolor{blue}{\emph{spectrum}} of the manifold.
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\item The spectrum determines many things about the space (like its volume). \item The spectrum determines many things about the space (like its volume).
\item When two distinct (up to isomorphism) manifolds share a spectrum, they for an \emph{isospectral pair}. \item When two distinct (up to isomorphism) manifolds share a spectrum, they for an \textcolor{red}{\emph{isospectral pair}}.
\end{enumerate} \end{enumerate}
\end{frame} \end{frame}

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\begin{frame}{Quotients of the $3$-sphere} \begin{frame}{Quotients of the $3$-sphere}
\begin{itemize} \begin{itemize}
\item $\so 4$ is isomorphic to the isometry group of $\S^3$. \item $\so 4$ is isomorphic to the isometry group of $\S^3$.
\item Subgroups $\Gamma \leqslant \so 4$ define equivalence classes of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^4$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$. \item Subgroups $\Gamma \leqslant \so 4$ define \textcolor{darkyellow}{equivalence classes} of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^4$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
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\item The obtain space is (\emph{sometimes}) a manifold. In particular, it is well defined and spherical for finite $\Gamma$. \item The obtain space is (\emph{sometimes}) a manifold. In particular, it is well defined and spherical for finite $\Gamma$.
\item This can be easily generalised to the $n$-sphere. \item This can be easily generalized to the $n$-sphere.
\end{itemize} \end{itemize}
\end{frame} \end{frame}