Let $M$ be a Riemannian smooth compact manifold.
It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ?


I've been looking for some references, but I can't find any dealing with this problem. I would appreciate if someone had an answer - even partial.



Triangulation with simplices of same volume
I don't know a reference, but I think that we can use a theorem of Moser to get what you want.


Start with any triangulation $\mathcal{T}$ of $M$, whose cardinal is denoted by $k$ and denote by $\omega$ the Riemannian volume form. There is a positive smooth function $f$ such that $\int\_M f\omega=\mathrm{vol}(M)$ (i.e. $f$ has average $1$) and such that for all $T\in\mathcal{T}$:
$$\int\_T f\omega = \frac{\mathrm{vol}(M)}{k}$$
(simply adjust $f$ in the interior of each simplex).


Now, [Moser proved in 1965](http://www.ams.org/journals/tran/1965-120-02/S0002-9947-1965-0182927-5/) (answering a question of a MO user by the way) that, since $f\omega$ and $\omega$ have the same total volume, there is a diffeomorphism $\phi:M\to M$ such that $\phi\_\*(f\omega)=\omega$. Then by the change of variable formula, 
$$\int\_{\phi(T)} 1\,\omega= \int\_T f\,\omega = \frac{\mathrm{vol}(M)}{k}$$
so that the collection $\{\phi(T)\}$ is a triangulation of $M$ with all simplices of the same volume.


