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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?

edited Jan 30 at 16:02

asked Jan 30 at 13:17

Benoît Kloeckner
5,489●18●29

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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 – Igor Belegradek Jan 30 at 14:13

@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. – Benoît Kloeckner Jan 30 at 16:02

Thanks also for the reference, I'll definitely have a look at it. – Benoît Kloeckner Jan 30 at 16:10

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@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. – Igor Belegradek Jan 30 at 17:10

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. – Sergei Ivanov Jan 31 at 10:01

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Alexandre Eremenko · Answer 1 · 2013-01-30 23:41:25Z UTC

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.

Smoothing of piecewise Euclidean Riemannian metrics

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