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I'm studying Tu's book Differential Geometry. Problem is

Two Riemannian manifolds $M$ and $M′$ of dimension 2 with a diffeomorphism $T:M→M′$ between them. For every point $p \in M$, there is a positive number $a(p)$ such that $$⟨T^∗(u),T^∗(v)⟩_{M′,F(p)}=a(p)⟨u,v⟩_{M,p}$$ for all $u,v\in T_pM$. Find the relationship between the Gaussian curvatures between the two manifolds.

I saw How Gaussian curvature is affected by a conformal map (using forms). But I cannot derive $\tilde\omega_2^1 = \omega_2^1 + \left(-\frac{\lambda_2}\lambda\theta^1 + \frac{\lambda_1}\lambda\theta^2\right)$ part in Ted Shifrin's answer. Can you explain it more specifically? Thank you.

probafds123
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  • Did you compute $d\tilde\theta^i$? – Ted Shifrin Nov 26 '20 at 04:13
  • I think that $d\tilde{\theta}^1 = d(\lambda \theta^1)$ ,but $\lambda$ is function of $p$ and $\theta$ is coframe, so $d(\lambda \theta^i) = d(\lambda) \wedge \theta^i - \lambda d(\theta^i)$ and $d(\lambda ) = \lambda_1 \theta^1 + \lambda_2 \theta^2$, so $d\tilde{\theta}^1 = (\lambda_1 \theta^1 + \lambda_2 \theta^2)\wedge \theta^1 - \lambda d(\theta^1)$. – probafds123 Nov 26 '20 at 04:21
  • IS this formula right? then $d\tilde{\theta}^1 = -\lambda_2 \theta^1 \wedge \theta^2 - \lambda d(\theta^1)$ is right? – probafds123 Nov 26 '20 at 04:23
  • No minus sign. OK, then go on and fit with the structure equations. – Ted Shifrin Nov 26 '20 at 04:24
  • @TedShifrin, $d\tilde{\theta}^1 = -\lambda_2 \theta^1 \wedge \theta^2 + \lambda d(\theta^1) = -\lambda \tilde{w_2^1}\wedge \theta^2$, and

    $d\tilde{\theta}^2 = \lambda_1 \theta^1 \wedge \theta^2 + \lambda d(\theta^2) = \lambda \tilde{w_2^1}\wedge \theta^1$

    Assume $\tilde{w_2^1} = a \theta^1 + b \theta^2$, then

    $-\lambda_2 \theta^1 \wedge \theta^2 + \lambda d(\theta^1) = - a \lambda \theta^1 \wedge \theta^2$.

    But I cannot get $a$ due to $d(\theta^1)$.

     – probafds123 Nov 26 '20 at 04:33
  • You haven't used the original structure equations, have you? – Ted Shifrin Nov 26 '20 at 04:47
  • @Ted Shifrin I think i used first structural equation. – probafds123 Nov 26 '20 at 04:50

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