Recently, I've been looking into the topic on quasi-local mass. This is a topic related to general relativity, and I'd like to acquaint myself with it via the book titled Geometric Relativity and written by Dan A. Lee. Here's a definition in which a statement puzzles me much.
Definition 6.1. Let $n\geq 3$, and let $(\Sigma^{n-1},\gamma)$ be a compact Riemannian manifold equipped with a nonnegative function $\eta$. We refer to the triple $(\Sigma,\gamma,\eta)$ as Bartnik data. We say that $(M^n,g)$ is an admissible extension of $(\Sigma,\gamma,\eta)$ if the following hold:
(1) $(M,g)$ is a complete, asymptotically flat manifold with boundary $\partial M$ identified with $\Sigma$.
(2) $R_g\geq 0$ in the interior of $M$.
(3) The metric on $\Sigma=\partial M$ induced by $g$ is equal to $\gamma$.
(4) Let $H_g$ denote the mean curvature of $\Sigma=\partial M$ in $(M,g)$, computed with respect to the "outward" normal $\nu$, which points into $M$. The mean curvature satisfies $H_g\leq\eta$.
(5) $\Sigma=\partial M$ is not enclosed by an apparent horizon (except in the case $\eta\equiv 0$, in which case we allow $\partial M$ itself to be a horizon).
In condition (1), what did the author mean by a manifold $M$ with boundary $\partial M$ identified with $\Sigma$? The author did mention $\Sigma=\partial M$ a couple of times in subsequent conditions, but I still hesitate about whether or not the symbol $=$ is the ordinary equal sign (like the one in high school math). Could anyone please give me a conclusive answer? Thank you.