Riemannian Geometry Part 5

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 $f_{*} R=\left.\bar{R}\right|_{f(M)}$.

Problem 6.141 Let $M$ be an $n$-dimensional $(n \geqslant 3)$, totally umbilical submanifold of a $2 m$-dimensional complex space form $(\widetilde{M}, g, J)$ of holomorphic sectional curvature $c \neq 0$. Prove that $M$ is one of the following submanifolds: (i) A complex space form holomorphically immersed in $\tilde{M}$ as a totally geodesic submanifold. (ii) A real space form (i.e. a not necessarily simply connected space of constant curvature) immersed in $\widetilde{M}$ as a totally real and totally geodesic submanifold. (iii) A real space form immersed in $\widetilde{M}$ as a totally real submanifold with nonvanishing parallel mean curvature vector. The relevant theory is developed, for instance, in Chen and Ogiue [7].

Problem 6.142 Consider the flat torus $T^{2}=\mathbb{R}^{2} / \Gamma$ defined by the lattice $$ \Gamma=\mathbb{Z} v_{1} \oplus \mathbb{Z} v_{2}, \quad v_{1}=(-\pi, \pi), v_{2}=(0,2 \pi) $$ (i) Prove that $$ f(u, v)=(\cos u \cos v, \cos u \sin v, \sin u \cos v, \sin u \sin v) $$ for $u, v \in(0,2 \pi)$ (see Remark 1.4), is an isometric embedding of $T^{2}$ into the unit sphere $S^{3}$ of $\mathbb{R}^{4}$. (ii) Prove that the total embedded curvature of $f\left(T^{2}\right)$ in $S^{3}$ is a constant. Hint Use the Generalised Gauss Theorema Egregium 6.30.