# Is a conformal transformation also a general coordinate transformation?

As far as I understand, a general coordinate transformation is induced by a diffeomorphism $f:M\rightarrow M$ where $M$ is a manifold (which can locally be described with coordinates). So if $x:M\rightarrow \mathbb{R}^m$ is a coordinate map, then $y:=x\circ f$ is a new coordinate map after the coordinate transformation. One could locally express the new coordinates in terms of the old ones, writing $y=y(x)$. Is that right so far?

Now if such a diffeomorphism is applied to all points, it should induce an effect on the metric. Let's say $X,Y$ are vector fields (elements of the module of derivatives over the ring of smooth functions $C^\infty(M)$ over $M$) and $g$ is the metric tensor field (an element of the comodule of the module of vector fields which takes two vector fields back to $C^\infty(M)$) and $p$ is a point of $M$. Then, is it right to say that the diffeomorphism induces the change \begin{equation} f^*g(X,Y)|_p = g_p(f_* X_p, f_* Y_p)? \end{equation} I have also seen in another post an action defined on all objects as in \begin{equation} (f^*)^{(-1)}g(f_* X,f_* Y)|_p = g_p(X_p, Y_p), \end{equation} which would mean that any diffeo would leave $g(X,Y)$ invariant? If this is correct, then I am confused here because I don't yet understand why the action should include the inverse of the pull-back?

A conformal transformation is defined as a diffeomorphism that leaves the metric invariant up to a an overall factor, meaning that the diffeomorphism induces a pull-back of the metric that is conformally equivalent (equivalent up to an overall factor) to the old one. Does this mean \begin{equation} f^*g(X,Y)|_p=g_p(f_* X_p, f_* Y_p)=\Omega(p) g_p(X_p,Y_p)? \end{equation}

In that case, a conformal transformation would be a coordinate transformation that changes the metric only by an overall factor and is thus also a coordinate transformation?

But then a conformal invariance of some theory would not be special anymore in a covariant formulation which confuses me. Thus my understanding of the action of a transformation on the metric and the vector fields is probably wrong at some (or multiple) point(s). Would be great if you could help me to clarify that.

• Does really no one have an answer? I thought it to be pretty clear for people studying differential geometry. Jan 28 '18 at 11:30
• I am also surprised there is not more interest here - I may work on a partial answer to spur activity. But I will point out that I don't think many people would agree with "A conformal transformation is defined as a diffeomorphism that leaves the metric invariant up to a an overall factor". I think the definition of a conformation transformation is simply $\Omega g$, purely a local map. May 24 at 19:32

A conformal transformation is a map that takes a metric $$g$$ (in a particular coordinate system) to $$\Omega g$$, but it's precise form depends on the local coordinate system. It actually can't be a coordinate system transform, because the scalar curvature is not invariant under a conformal transformation.