Metric for subset of $\mathbb{R}^n$

Question

Consider the collection of points in $\mathbb{R}^n$ whose coordinates are all strictly positive. I want to think of this subset as a Riemannian manifold. Does anyone happen to know what the Reimannian metric would be?

Answer 1

The set $P = \{(x_1, \dots, x_n) \in \mathbb{R}^n \mid x_1, \dots, x_n > 0\}$ is open and is therefore an open submanifold of $\mathbb{R}^n$.

The restriction of a Riemannian metric to an open submanifold is again a Riemannian metric. Therefore $P$ carries a natural Riemannian metric given by $g|_P$ where $g = dx^1\otimes dx^1 + \dots + dx^n\otimes dx^n$ is the usual Riemannian metric on $\mathbb{R}^n$.