# Recovering metric from Laplace-Beltrami operator

On a Riemannian manifold $$(M, g)$$, if one is given an oracle $$O$$ that allows one to evaluate the Laplace-Beltrami operator, how can we recover the metric $$g$$? More precisely, $$O: (M \rightarrow \mathbb R) \rightarrow \mathbb R$$, $$O(f) \equiv \Delta f = \frac{1}{\sqrt{|g|}} \partial_i (\sqrt{|g|} g^{ij} \partial_j f)$$. We are allowed to pick any number of functions $$f$$ to evaluate against the oracle $$O$$. We do not know the symbolic expression for either $$g$$ or $$\Delta$$.

As far as I can tell, the problem is by by using $$O$$, we can only access the components of the metric under a $$\partial_i$$. I'm at a loss on how to "recover" the metric $$g$$ from $$\partial_i (\cdots g^{ij} \cdots)$$.

• Very good question. It received a great answer. I learned something from both. Sep 22, 2020 at 10:49

## 1 Answer

Given a point $$p\in M$$, you can recover the metric at $$p$$ as follows. Choose any local coordinates $$(x^1,\dots,x^n)$$ such that $$p$$ has the coordinate representation $$(0,\dots,0)$$. For any indices $$k,l$$, let $$f_{kl}(x) = \tfrac 1 2 x^k x^l$$. If you expand out the expression for $$\Delta (f_{kl})$$ and use the fact that $$x^k=x^l=0$$ at $$p$$, you'll find that $$\Delta(f_{kl})(p) = g^{kl}$$. Then you can use matrix inversion to recover $$g_{kl}$$.

If your oracle is restricted to acting only on globally defined functions, you can use a bump function to extend the coordinate functions to global smooth functions $$(u^1,\dots,u^n)$$ that agree with $$(x^1,\dots,x^n)$$ in a neighborhood of $$p$$, and let $$f_{kl}(x) = u^k u^l$$.

This is a special case of a much more general construction: if $$P$$ is an $$m$$th order scalar linear partial differential operator, the coefficients of its highest-order terms determine a coordinate-independent symmetric contravariant $$m$$-tensor field called the principal symbol, which can be evaluated at any point by applying $$P$$ to $$m$$-fold products of coordinate functions. The principal symbol of $$\Delta$$ is the associated cometric, which is the induced metric on $$T^*M$$.

• Thank you for this wonderful answer that both answers and teaches :) If I wanted to learn more about principal symbols, what book would I read, as an undegraduate? Also, thanks for your great book! Dec 27, 2019 at 15:49
• @SiddharthBhat: I don't know of any undergraduate texts. But you can find a good description in the first volume of Michael Taylor's Partial Differential Equations series. Dec 27, 2019 at 16:24