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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.

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If $A$ is the invertible matrix that sends each $e_i$ to $f_i$, then the the matrix that sends $(e_1,\dots,e_n,u)$ to $(f_1,\dots,f_n,u)$ has block form $\begin{pmatrix} A& 0 \\ 0 & 1\end{pmatrix}$ so its determinant equals the determinant of $A$. Hence your assumptions imply that also $(f_1,\dots,f_n,u)$ is positively oriented.

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