# Extending to a local frame that agrees with given orientation

Suppose that $$(e_1, \ldots, e_k)$$ is an oriented basis for $$T_pM$$ where $$M$$ is an oriented Riemannian manifold. In general, we know that we can extend to a smooth local frame $$(X_1, \ldots, X_k)$$ on $$U\ni p$$ such that $$X_i\rvert_p = e_i$$ for each $$i$$.

But can we further stipulate that $$(X_1\rvert_q, \ldots, X_k\rvert_q)$$ is oriented for each $$q\in U$$?

I have tried thinking of ways to shrink $$U$$ in a suitable way. I have played around with using the orientation form $$\omega$$ on $$M$$ and with first applying Gram-Schmidt to $$(X_1, \ldots, X_k)$$, but no progress.

Any help is much appreciated.

The local frame $$(X_i)$$ will agree with the chosen orientation form: if $$\omega(e_1,\dots,e_k)>0$$, then $$\omega(X_1,\dots,X_k)>0$$ in $$U$$, because $$\omega(X_1,\dots,X_k)|_q=0$$ would imply that the $$X_i's$$ do not form a basis of $$T_qM$$.
• This solution shows how to restrict $U$ appropriately, but I don't see why vanishing at $\omega \rvert_q$ would imply that the $X_i$ don't form a basis. It's possible that an alternating tensor vanishes at a linearly independent tuple, right? – CuriousKid7 Dec 5 '18 at 5:01
• You don't need to restrict $U$ in any way. $M$ is orientable and you have fixed an orientation form $\omega$. What I'm saying is that if $(X_i)$ is a frame inside $U$, then $\omega(X_1,\dots,X_k)$ doesn't change sign inside $U$, because if it does there is a point at which it must vanish, and at that point $(X_i)$ would not be a frame – Federico Dec 5 '18 at 13:51
• "It's possible that an alternating tensor vanishes at a linearly independent tuple, right?" No, unless $\omega=0$ at that point. If it vanishes at a linearly independent tuple, it vanishes on any tuple. – Federico Dec 5 '18 at 13:52
• Assume $\omega(e_1,\dots,e_k)=0$ for some basis $(e_i)$. Take any $k$-uple $v_j=a^i_je_i$. Then $\omega(v_1,\dots,v_k)=\det(A)\omega(e_1,\dots,e_k)=0$, where $A$ is the matrix with entries $a^i_j$. – Federico Dec 5 '18 at 13:54