Why Riemannian metric does not depend on the choice of coordinate system

proof-verificationriemannian-geometrysmooth-manifolds

In DoCarmo’s Riemannian Geometry book, a Riemannian metric on a smooth manifold $M$ is defined to be a correspondence which associates to each point $p$ of $M$ an inner product $<,>_p$ on the tangent space $T_pM$ such that if $x:U\subset\mathbb R^n\rightarrow M$ is a coordinate system around $p$, with $x(x_1,…,x_n)=q\in x(U)$, then $g_{i,j}(x_1,…,x_n):=<dx_q(e_i),dx_q(e_j)>_q$ is a smooth function on $U$.

Now I want to prove that this definition does not depend on the choice of coordinate system. So suppose $y:V\rightarrow M$ is another coordinate system around $p$. We have $<dy_q(e_i),dy_q(e_j)>_q=<dx_q(d(x^{-1}y)_q(e_i)),dx_q(d(x^{-1}y)_q(e_j))>_q$, and since the terms $d(x^{-1}y)_q(e_i)$ and $d(x^{-1}y)_q(e_j)$ are smooth linear combinations of the standard bases of $\mathbb R^n$ as a map of $q$, using the bilinearity of the inner product and our hypothesis about the coordinate system $x$, we deduce the same for the coordinate system $y$.

Is my reasoning correct?

Best Answer

I think you have not intepreted the question correctly (or at least you have left too many words out that I cannot tell whether you have interpreted it correctly).

In particular, I do not see that you have actually used the definition of a smooth manifold, in particular you have not made use of the key fact about smooth manifolds, namely smoothness of overlap maps. You write about two coordinate systems $x$ and $y$, and you mention the overlap $x^{-1} y$, but you never mention or make use of the key fact that the overlap maps $x^{-1} y$ and $y^{-1} x$ are smooth.

Let me separate the definition of a Riemannian metric on $M$ into two parts:

  1. The object given;
  2. The property that object must satisfy.

The object given is:

The correspondence which associates to each point $p \in M$ an inner product $\langle,\rangle_p$ on the tangent space $T_p M$".

Up to here, coordinate systems are not involved, but now they come into the property that the correspondence must satisfy:

If $x:U\subset\mathbb R^n\rightarrow M$ is a coordinate system around $p$, with $x(x_1,...,x_n)=q\in x(U)$, then $g_{i,j}(x_1,...,x_n):=<dx_q(e_i),dx_q(e_j)>_q$ is a smooth function on $U$.

The issue is whether this property depends on the coordinate system.

What one must show is if $y:V\rightarrow M$ is another coordinate system around $p$, and if we let $h_{i,j}(y_1,...,y_n) := \langle dy_q(e_i),dy_q(e_j) \rangle_q$, then on the overlap $U \cap V$, to say that each of the functions $g_{i,j} \mid x(U \cap V)$ is smooth is equivalent to saying that each of the functions $h_{i,j} \mid y(U \cap V)$ is smooth.

The way you prove this equivalence is to write out how to relate the maps $g_{i,j} \mid x(U \cap V)$ and $h_{i,j} \mid y(U \cap V)$ (for each $i,j$) are related, expressing this relation as a equation obtained from the chain rule, involving the Jacobian matrices for the overlap maps $x^{-1} y$ and $y^{-1} x$.

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