Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on $M$, which is singular on (part of) the codimension $2$ skeleton of $T$.

Is it possible to approximate $g_0$ by a smooth Riemannian metric? The approximation should in particular change length of curves and the volume by arbitrarily small amounts.

I guess the answer is positive and well-known, but I did manage to find a reference (in particular, several works ask the smoothing to satisfy certain curvature assumptions, which I do not). Is there a reference or are there obstruction to smoothing?

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    $\begingroup$ You need to assume that $M$ is smooth and $T$ is a smooth triangulation: not every PL manifold has a smooth structure. As for your question, Ontaneda had to address similar issues in sections 7-8 of front.math.ucdavis.edu/1110.6374 $\endgroup$ – Igor Belegradek Jan 30 '13 at 14:13
  • $\begingroup$ @Igor Belegradek: thanks, I did thought about smooth manifold only, but forgot to write it. Now edited. As for the triangulation, any assumption that does not prevent it to exist is fine. $\endgroup$ – Benoît Kloeckner Jan 30 '13 at 16:02
  • $\begingroup$ Thanks also for the reference, I'll definitely have a look at it. $\endgroup$ – Benoît Kloeckner Jan 30 '13 at 16:10
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    $\begingroup$ @Benoît, what I meant is even if $M$ is smooth it may be that your triangulation is non-smoothable or has a smoothing of $M$ that is different from the original one. Thurston in [Three-dimensional geometry and topology, Princeton Mathematical Series, 35] discusses the issue (around page 197, if memory serves) and sketches that any low-dimensional PL manifold is smoothable by a direct geometric argument. This would be a starting point, and then one would have smooth the metric, but without matching the smoothing with the original smooth structure on $M$ there is no way to proceed. $\endgroup$ – Igor Belegradek Jan 30 '13 at 17:10
  • $\begingroup$ You cannot change lengths by arbitrarily small amounts, even in the 2-dimensional case. Consider a cone singularity and a loop near the apex wrapping many times around it. $\endgroup$ – Sergei Ivanov Jan 31 '13 at 10:01

Such results are known for surfaces. I believe the correct reference is MR0126778 Rešetnjak, Ju. G. Isothermal coordinates on manifolds of bounded curvature. I, II. (Russian) Sibirsk. Mat. Ž. 1 1960 88–116, 248–276.

There are also two books of A. D. Aleksandrov and V. A. Zalgaller, and more recent survey of Reshetnyak in the Springer Encyclopaedia MR1263963.


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