Let $\{v_1,\cdots,v_k\}$ be the vertices of a regular $(k-1)$-simplex $\Delta(k,\ell)$, with a given metric such that the pairwise distance between the vertices is $\ell$.

Given a Riemannian manifold $M^n$. Is there any criterion such that $\Delta(k,\ell)$ can be isometrically embedded in $M$(i.e. preserve the distance, here we only consider the embedding of the vertexes and 1-skeleton).

For example $\Delta(3, \ell)$ can be isometrically embedded into $S^2$ for $\ell$ not too large. $\Delta(4,\ell)$ can only be embedded into $S^2$ with $\ell=\arccos(-1/3)$. What can we say about $\mathbb{CP}^n$ or other symmetric spaces?