I am so confused about the following problem (maybe I have made some silly conceptual mistakes)
The reference I used is Lee's introduction to Riemannian manifold.
There are two theorems in this book.
According to my understanding, the sectional curvature is a scalar, which can be computed by any basis of the tangent space by formula in theorem 8.29 (i.e. does not depend on the basis). Now, in exercise 8.30, $\lambda$ is a constant, substitute it into theorem 7.30 shows $f$ is a constant.
Then it seems (7.44) and (7.46) give different answers. 7.44 and 8.29 implies $\tilde{sec}(\Pi)=\lambda sec(\pi)$ while (7.46) implies $\tilde{sec}(\Pi)=\lambda^{-1} sec(\pi)$. How can this happen?
Does anyone have any ideas or comments?
Applying $(7.44)$ with constant $f$ shows that $\widetilde{\text{Rm}}=\lambda\cdot\text{Rm}$. So, by using formula $(8.28)$, we get that for a plane $\Pi$ spanned by two linearly independent vectors $v,w$, \begin{align} \widetilde{\text{sec}}(\Pi)&=\frac{\widetilde{\text{Rm}}(v,w,w,v)}{|v\wedge w|_{\widetilde{g}}^2}=\frac{\lambda\cdot \text{Rm}(v,w,w,v)}{\lambda^2\cdot|v\wedge w|_g^2}=\frac{1}{\lambda}\cdot\text{sec}(\Pi), \end{align} which is exactly what is claimed. I think you forgot to account for the different norm in the denominator.