= Solution
On $S^n\times S^n$, consider $q(x,y)=\langle x,y\rangle$. At a point of $q^{-1}(0)$, its differential is
$$
dq_{(x,y)}(u,v)=\langle u,y\rangle+\langle x,v\rangle.
$$
The tangent vector $(y,x)$ lies in $T_xS^n\oplus T_yS^n$ and has $dq(y,x)=2$, so zero is a <regular value>. The <regular level set theorem> makes $P=q^{-1}(0)$ a smooth submanifold. Its tangent space is
$$
T_{(x,y)}P=\left\{(u,v):
\langle u,x\rangle=0,
\ \langle v,y\rangle=0,
\ \langle u,y\rangle+\langle x,v\rangle=0
\right\}.
$$
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