ALIBABA 2021 Problem 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 $n=2$ .