Suppose that $S$ is a smooth $n$-submanifold of $M$ where $\dim(M) = n+1$. Suppose also that $S$ and $M$ are both Riemannian and oriented.
Suppose that $(e_1, \ldots, e_n)$ and $(f_1, \ldots, f_n)$ are both positively oriented orthonormal bases for $T_pS$ and that $(e_1, \ldots, e_n, u)$ is positively oriented for $T_pM$ where $u$ is orthonormal with respect to $(e_1, \ldots, e_n)$.
Is it necessarily true that $(f_1, \ldots, f_n, u)$ is also positively oriented?
It seems to me possible that $(f_1, \ldots, f_n, -u)$ is positively oriented. But I am reading a proof that seems to be relying on the fact that it cannot be.
Any help is appreciated.