Let $\mathcal{M}$ be the convex cone of symmetric positive definite $n\times n$ real matrices. $\mathcal{M}$ is an $\frac{n(n+1)}{2}$-dimenasional Riemannian manifold.
Could you help me proving (or disproving) that $\mathcal{M}$ is isometric (in the Riemannian sense) to some Euclidean space $\Bbb{R}^n$?
Excuse my ignorance, please help me figuring that out! Thanks a lot!