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Sep 6, 2018 at 18:52 comment added Vivek Shende Dmitri’s comment answered my question. I left it in its original form in case anyone else has the same one.
Sep 6, 2018 at 15:37 comment added Paul 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?
Sep 4, 2018 at 2:23 history edited Vivek Shende CC BY-SA 4.0
added 46 characters in body
Sep 4, 2018 at 1:31 comment added Dmitri Pavlov 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.
Sep 3, 2018 at 5:02 history edited Vivek Shende CC BY-SA 4.0
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Sep 3, 2018 at 4:48 history asked Vivek Shende CC BY-SA 4.0