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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?

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    $\begingroup$ Your definition of smooth triangulations is different from the standard one (see Munkres, Elementary differential topology, Definition 8.3). Using the standard definition, the answer to your question is positive and follows immediately from the definition. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 4, 2018 at 1:31
  • $\begingroup$ After Dimitri's comment the answer to your question is still yes. Another question would be if there are some topological manifolds that admit a triangulation in the first sense but not in the second sense. Is this your question now or I'm missing something? $\endgroup$
    – Paul
    Commented Sep 6, 2018 at 15:37
  • $\begingroup$ Dmitri’s comment answered my question. I left it in its original form in case anyone else has the same one. $\endgroup$
    – Vivek Shende
    Commented Sep 6, 2018 at 18:52

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