I am reading "Analysis on Manifolds" by James R. Munkres.
In $\mathbb{R}^3$, a frame $(a_1,a_2,a_3)$ is right-handed if curling the fingers of one's right hand in the direction from $a_1$ to $a_2$ makes one's thumb point in the direction of $a_3$. See Figure 20.2.
One way to justify this statement is to note that if one has a frame $(a_1(t),a_2(t), a_3(t))$ that varies continuously as a function of $t$ for $0\leq t\leq 1$, and if the frame is right-handed when $t=0$, then it remains right-handed for all $t$. For the function $\det [a_1\, a_2\, a_3]$ cannot change sign, by the intermediate-value theorem. Then since the frame $(e_1,e_2,e_3)$ satisfies the "curled right-hand rule" as well as the condition $\det [e_1\, e_2\, e_3]>0$, so does the frame corresponding to any other position of the "curled right hand" in $3$-dimensional space.
Let $(a_1,\dots,a_n)$ be a right-handed frame in $\mathbb{R}^n$.
Let $(b_1,\dots,b_n)$ be a right-handed frame in $\mathbb{R}^n$.
Is there a frame $(c_1(t),\dots,c_n(t))$ that varies continuously as a function of $t$ for $0\leq t\leq 1$ and that satisfies $(c_1(0),\dots,c_n(0))=(a_1,\dots,a_n)$ and $(c_1(1),\dots,c_n(1))=(b_1,\dots,b_n)$?
If so, how to prove this?
