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Consider the local representation of the Laplace Beltrami operator on a Riemannian n - dimensional manifold $(M,g)$: \begin{equation} \triangle_g = \frac{1}{\sqrt{\text{det}(g)}} \sum^n_{i,j = 1} \frac{\partial}{\partial x^i} g^{ij} \sqrt{\text{det}(g)}\frac{\partial}{\partial x^j} \end{equation}

Some time ago I read in some textbook that one can, locally at any point $p \in M$, choose a small enough neigborhood $U \subset M$ so that by a linear transformation of the coordinates the matrix representation of $g$ in $U$ is $g_{ij} = \delta_{ij}$.

But wouldn't this imply that the above expression always simplifies to the expression \begin{equation} \triangle_g = \sum^n_{i = 1} \frac{\partial^2}{\partial (x^i)^2} \quad ? \end{equation}

Alternatively I would think that even though the metric evaluates to the Kronecker delta that doesn't mean that its derivatives are zero. So do we actually then have

\begin{equation} \triangle_g = \sum^n_{j = 1} \frac{\partial^2}{\partial (x^j)^2} + \sum_{j = 1}^n ( \frac{\partial}{\partial x^j} \sqrt{\text{det}(g)} + \sum_{i = 1}^n \frac{\partial}{\partial x^i} g^{ij})\frac{\partial}{\partial x^j} \end{equation}

I am sure at least one my impression is false, if anybody could help pointing out what I got wrong, or refer to a reference where I could read about diagonalization of metrics that would be so helpful, many thanks !

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Your claim is false: if there were a choice of local coordinates so that $g_{ij} = \delta_{ij}$ then the manifold would be flat in that neighbourhood, but flatness is a isometry invariant. What is true is that there are always local coordinates so that the metric at a point has components $\delta_{ij}$. –  Zhen Lin Jan 21 '12 at 1:17
    
ok, so with the edit that I just posted, would that then account for the fact that, pointwise, the metric is $g_{ij} = \delta_{ij}$ though I have variation as I travel away from the point ? Thanks for your help! –  harlekin Jan 21 '12 at 1:29
    
That makes no sense at all. $\delta_{ij}$ is a constant – how can there be any variation? –  Zhen Lin Jan 21 '12 at 1:39
    
what I mean by that is at the point $p$ the matrix that represents the inner product on the tangent space $T_pM$ is the Identity, i.e. the metric $g$ evaluated at $p$ is equal to $M$. But as you rightly say the metric is only $\delta{ij}$ at that point, it is not constant on the whole neighborhood. Is that correct ? –  harlekin Jan 21 '12 at 1:53

2 Answers 2

up vote 4 down vote accepted

In fact, the following is true: At any point $p$, there exists a neighborhood $U$ of $p$ such that $g_{ij}=\delta_{ij}$, the Kronecker delta, and $\displaystyle\frac{\partial g_{ij}}{\partial x_k}$ at $p$ is zero. (It's important to keep in mind that $\displaystyle\frac{\partial g_{ij}}{\partial x_k}$ only vanishes at $p$, not in a neighborhood; otherwise, it has constant curvature.) If I remember correctly, this is called normal coordinates. If you look at the book of Riemannian Geometry of Do Carmo, you need to prove the existence of normal coordinates in an exercise. Maybe in other books they have the proof.

Anyway, in the normal coordinates, as you have calculated, we have \begin{equation} \triangle_g = \sum^n_{j = 1} \frac{\partial^2}{\partial (x^j)^2} + \sum_{j = 1}^n ( \frac{\partial}{\partial x^j} \sqrt{\text{det}(g)} + \sum_{i = 1}^n \frac{\partial}{\partial x^i} g^{ij})\frac{\partial}{\partial x^j}. \end{equation} Because $\displaystyle\frac{\partial g_{ij}}{\partial x_k}=0$ at $p$, the second term vanishes. So at the point $p$, the Laplace Beltrami operator is given by \begin{equation} \triangle_g = \sum^n_{i = 1} \frac{\partial^2}{\partial (x^i)^2}. \end{equation}

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Many thanks for your answer, that helped a lot !! –  harlekin Jan 21 '12 at 10:42
    
@harlekin: You are welcome. Good that it helps. –  Paul Jan 21 '12 at 11:01

This question is partly about the metric and how to diagonalize it, so let me focus on this. I think some amphiboly is present here: one must be careful to distinguish between coordinates and coordinates.

By "normal coordinates", one means really, choose an orthonormal basis for the tangent bundle over a neighborhood about $p$. As Zhen Lin points out, the obstruction to integrating this basis to coordinates is curvature. Such an orthonormal basis is also called a tetrad or a verbein.

Contrast these normal "coordinates" to a coordinate system about $p$ in the sense of a smooth manifold, a homeomorphism between a neighborhood about $p$ to $\mathbb{R}^n$. Coordinate vector fields correspond to the images of the coordinate vector fields in $\mathbb{R}^n$ under this map. Such fields are of course a local basis for $TM$, but not all bases for $TM$ can be integrated to coordinates.

So "coordinates" means two different things: "local basis for $TM$" if you're talking about tensor quantities, or "local homeomorphism to $\mathbb{R}^n$" otherwise.

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Very interesting, that helps to better understand the general idea behind it, tks for pointing this out! –  harlekin Jan 21 '12 at 10:38
    
Glad to help :) –  Neal Jan 23 '12 at 2:44

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