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On the three-sphere $S^3$, I'm given three vector fields $X$, $Y$ and $Z$, such that at each point $p\in S^3$, the tangent vectors $X_p$, $Y_p$ and $Z_p$ form an orthogonal basis of the tangent space $T_pS^3$.

I denote by $X^t:S^3\rightarrow S^3$ the flow of the vector field $X$, so $X^0(p)=p$ and $\frac{d}{dt}X^t(p)=X_{X^t(p)}$.

Its pushforward at the point $p$ is denoted $(X^t)_{*,p}$, so this is a map from $T_pS^3$ to $T_{X^t(p)}S^3$.

Now let $v$ be an arbitrary tangent vector in $T_pS^3$. We can write $v=v_xX_p+v_yY_p+v_zZ_p$. I want to prove the following:

$(X^t)_{*,p}(v)=v_x X_{X^t(p)}+...$.

Here the dots denote other terms in the span of $Y_{X^t(p)}$ and $Z_{X^t(p)}$, but it's only the $X_{X^t(p)}$ component I care about (plus I don't know the other components).

Is this maybe immediate from the definition of the flow? I tried using the defining equation $\frac{d}{dt}X^t(p)=X_{X^t(p)}$ of the flow of $X$, but I don't know how I can relate the time derivative with the pushforward...

Edit: so I think I can visualize my question as follows. Starting with a tangent vector $v$ at $p$, flowing this vector along $X$ (using the pushforward of the flow) does not change the component of $v$ along $X$. Sounds plausible...

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