Problem
Problem 6.140 Prove:
1. Every strictly conformal map is an immersion.
2. If $M$ is connected, then a strictly conformal map
$$
f:(M, g) \rightarrow(\bar{M}, \bar{g})
$$
of ratio $\lambda$ transforms the Levi-Civita connection $\nabla$ of $g$ into the Levi-Civita connection $\bar{\nabla}$ of $\bar{g}$, if and only if $\lambda=$ const and the second fundamental form of the immersed submanifold $f(M)$ vanishes.
3. If $\lambda=1$, that is, $f$ is an isometry, and the second fundamental form of $f(M)$ vanishes, then if $R$ and $\bar{R}$ stand for the Riemann-Christoffel curvature tensors of $M$ and $\bar{M}$, respectively, one has $f_{*} R=\left.\bar{R}\right|_{f(M)}$.