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Found problems: 1

2021 Alibaba Global Math Competition, 13
Let $M_n=\{(u,v) \in S^n \times S^n: u \cdot v=0\}$, where $n \ge 2$, and $u \cdot v$ is the Euclidean inner product of $u$ and $v$. Suppose that the topology of $M_n$ is induces from $S^n \times S^n$. (1) Prove that $M_n$ is a connected regular submanifold of $S^n \times S^n$. (2) $M_n$ is Lie Group if and only if $n=2$.
group theory manifold submanifold topology geometry