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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.

gžd15
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    https://math.stackexchange.com/questions/98113/conformal-transformation-of-the-curvature-and-related-quantities this should help – Frieder Jäckel Aug 22 '18 at 11:30
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    https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry#Under_a_conformal_change – Travis Willse Aug 23 '18 at 14:20
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    If you think about the diffeomorphism as an identification of the two manifolds, it will greatly simplify the notation (Hint: the metric just gets multiplied by a strictly positive scalar function). – Yuri Vyatkin Aug 25 '18 at 01:49

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