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Let $\rho_t$ be the one-parameter flow associated to a vector field $v$ on a manifold $M$.

What is $\frac{d}{dt} d_p\rho_t$?

Intuitively, $$ \frac{d}{dt} d_p\rho_t = d_p \frac{d}{dt} \rho_t = d_p (v \circ \rho_t) = d_{\rho_t(p)}v \cdot d_p \rho_t. $$

However, this seems wrong as $v$ is a vector field and $dv$ appears.

Any help or clarification would be greatly appreciated.

mathandchess
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  • If $M$ is not Euclidean space, please identify the spaces the quantities live in. – Ted Shifrin Jul 17 '22 at 03:39
  • If $\rho_t$ is a family of diffeomorphisms $M\to M$, then $d\rho_t$ should be a family of diffeomorphisms $TM\to TM$, and hence its time derivative should be a vector field on $TM$, no? – Jackozee Hakkiuz Jul 17 '22 at 03:55
  • Oh but $d_p\rho_t$ is a map $T_pM\to T_{\rho_t(p)}M$. How do you differentiate this family of maps with respect to $t$? each has a different codomain. – Jackozee Hakkiuz Jul 17 '22 at 04:37
  • Unless you mean the derivative $d/dt$ to act at the end, after $d_p\rho$ has already acted on a tantent vector at $p$ and the result has acted on a smooth function? – Jackozee Hakkiuz Jul 17 '22 at 04:39
  • In what context did you find this object? – Jackozee Hakkiuz Jul 17 '22 at 05:28
  • @JackozeeHakkiuz I'm actually after $\frac{d}{dt} [(d\rho_t)^{-1} (w\circ \rho_t)] ,$ where $w$ is another vector field on $M$. – mathandchess Jul 17 '22 at 21:42
  • @mathandchess and what object do you expect it to be? $(d\rho_t)^{-1}(w\circ\rho_t)$ is a vector field for each $t$. So $\frac{d}{dt}[(d\rho_t)^{-1}(w\circ\rho_t)]$ is also a vector field for each $t$. For $t=0$, you get the definition of the Lie derivative $L_vw$. Is that what you want? – Jackozee Hakkiuz Jul 17 '22 at 22:50
  • @JackozeeHakkiuz I guess I'm trying to show directly that this expression is equivalent to the definition via the commutator of vector fields. – mathandchess Jul 17 '22 at 23:41

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