Visualizing / Interpreting Hessian Manifold in Mirror Descent

Question

Consider a Riemannian manifold which is some convex subset $K$ of $\mathbb{R}^d$ equipped with a metric that is the Hessian of a strictly convex function $\varphi:K \to \mathbb{R}$, i.e. $\langle u,v \rangle_x = \langle u, \nabla^2\varphi(x) \,v\rangle$. Gradient flow on such a manifold is the limit of mirror descent with Bregman divergence determined by $\varphi$, hence my interest. Is there a straightforward way to interpret the geometry of this manifold in terms of $\varphi$? A common example is $K$ being the probability simplex and $\varphi$ being the negative entropy function - is there a canonical Euclidean embedding of this manifold?