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Add figure on boundary orientation
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Signed-off-by: Marcello Seri <[email protected]>
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mseri committed Jan 19, 2022
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Expand Up @@ -277,6 +277,9 @@ \section{Orientation on manifolds}

Let's first look at the tangent space.
If $M$ ia a smooth $n$-manifold with boundary and $p\in \partial M$, we have three types of possible vectors:
\begin{marginfigure}
\includegraphics{images/8_2-boundary-vectors.pdf}
\end{marginfigure}
\begin{enumerate}
\item tangent boundary vectors: $X\in T_p(\partial M)\subset T_p M$ tangent to the boundary, forming an $(n-1)$-dimensional subspace of $T_p M$;
\item inward pointing vectors: $X\in T_pM$ such that $X = \varphi^{-1}_*(Y)$ where $\varphi^{-1}: V\subset \cH^n \to M$ and $Y$ is some vector $Y = (Y_1, \ldots, Y_n)$ with $Y_n > 0$;
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