In the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a smooth map from a neighborhood of the standard simplex in $\mathbb{R}^n$. Smooth manifolds always have them.

However, one doesn't really understand, in the image of such a simplex, the smooth geometry of how the various faces meet each other. If you wanted this, you would ask for more: that the map on each simplex can be extended to a diffeomorphism from a neighborhood of the standard simplex, to a neighborhood of its image in the manifold.

Are there such triangulations of smooth manifolds?

In (as it turned out my misunderstanding of) the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a smooth map from a neighborhood of the standard simplex in $\mathbb{R}^n$. Smooth manifolds always have them.

However, one doesn't really understand, in the image of such a simplex, the smooth geometry of how the various faces meet each other. If you wanted this, you would ask for more: that the map on each simplex can be extended to a diffeomorphism from a neighborhood of the standard simplex, to a neighborhood of its image in the manifold.

Are there such triangulations of smooth manifolds?

In the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a smooth map from a neighborhood of the standard simplex. Smooth manifolds always have them.

One could ask for more: that the map on each simplex can be extended to a diffeomorphism from a neighborhood of the standard simplex.

Are there non-cuspy triangulations of smooth manifolds?

In the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a smooth map from a neighborhood of the standard simplex in $\mathbb{R}^n$. Smooth manifolds always have them.

However, one doesn't really understand, in the image of such a simplex, the smooth geometry of how the various faces meet each other. If you wanted this, you would ask for more: that the map on each simplex can be extended to a diffeomorphism from a neighborhood of the standard simplex, to a neighborhood of its image in the manifold.

Are there such triangulations of smooth manifolds?