Minicursos

Minicurso

Título: Isometric Immersions of $\mathbb S^n$ into $\mathbb S^m$

Inscrição: aqui

Professor: Guilherme Machado de Freitas (IME)

Data: 18 a 22 de janeiro

Hora: 10h às 12h

Link: https://meet.google.com/yhw-pirn-zxy

Resumo: The Killing-Hopf theorem states that space forms are the only complete simply connected homogeneous isotropic spaces, which makes them the most natural ambient spaces for studying submanifolds. Moreover, by rescaling the Riemannian metric, one can reduce any space form $\mathbb{Q}^m_c$ to Euclidean space $\mathbb{R}^m$, the sphere $\mathbb{S}^m$ or the hyperbolic space $\mathbb{H}^m$, according to whether $c=0$, $c>0$ or $c<0$, respectively.

The simplest examples of submanifolds in space forms are the totally geodesic ones. The Chern-Kuiper inequality together with the completeness of the minimum relative nullity foliation leads to the fact that every isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^m$ is totally geodesic if $m<2n$, and the same is not true of $\mathbb{R}^m$ and $\mathbb{H}^m$ even in the hypersurface case $m=n+1$. On the other hand, the round sphere in the universal covering of the Clifford torus $\mathbb{T}^{n+1}$ provides an example of a nontotally geodesic isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^{2n+1}$. The existence of nontotally geodesic isometric immersions of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$ remains a major open problem in the field of submanifold geometry for $n>2$. A positive answer for dimension $n=2$ was obtained by Ferus and Pinkall in 1987. The study of isometric immersions of $\mathbb{S}^n$ into $\mathbb{S}^m$ is the main goal of this summer course.