This is an exercise from GTM 275, Differential Geometry by Loring W. Tu, page 94.
Let $T:M \rightarrow M'$ be a diffeomorphism of Riemannian manifolds of dimension 2. Suppose at each point $p \in M$, there is a positive number $a(p)$ such that
$${\left\langle {{T_ * }v,{T_ * }w} \right\rangle _{M',T(p)}} = a(p){\left\langle {v,w} \right\rangle _{M,p}}$$
for all $u,v \in T_p(M)$. Find the relation between the Gaussian curvatures of $M$ and $M'$.
In this book, the Gaussian curvature $K$ at a point $p$ of a Riemannian 2-manifold $M$ is defined to be
$${K_p} = \left\langle {{R_p}(u,v)v,u} \right\rangle $$
for any orthonormal basis $u,v$ for the tangent plane $T_pM$.
I'm stuck on how to express the relation between those quantities on $M$ and $M'$ by $T_*$ and $T^*$ without the viewpoint of identifying them as one manifold.
Any help will be appreciated.