\documentclass{article}
\usepackage[utf8]{inputenc}
\begin{document}

Let $P$ and $Q$ denote two meshes from two successive time steps. Without loss of generality, let $P$ denote the lower mesh and $Q$ denote the upper mesh. Then we use $N_P$ and $N_Q$ denote the sets of all nodes in $P$ and $Q$, respectively.

\textbf{Definition 1} (Local) spacetime. A \textit{spacetime} is given by two meshes, $P$ and $Q$, of two successive time steps, and mapping (connectivity) $N:N_P\rightarrow N_Q$ such that any $p\in N_P$ is mapped to a $q\in N_Q$. (note: injection here. To be more general: if $N(p)$ exist, then $N(p)\in N_Q$.

\textit{Local spacetime} is a subset of spacetime.

\textbf{Definition 2} Partition of (local) spacetime.

\textbf{Definition 3} Cutting faces. A cutting face is created by an edge in lower mesh that maps to an edge in upper mesh. It is composed of four edges: an edge in lower mesh, an edge in upper mesh, and two edges for node-to-next-node connectivities.

\textbf{Definition 4} Cutting paths. A cutting path is a closed path generated by a series of cutting faces.

Cutting paths are paired.
Cutting paths in lower and upper meshes are isomorphic.

\textbf{Lemma 1} 

\textbf{Lemma 2}

\end{document}
