I am reading Peter Petersen's notes on manifold theory and he introduces Lie Derivatives in the following way.

"Let $X$ be a vector field and $F^t$ the corresponding locally defined flow on a smooth manifold $M$. Thus $F^t (p)$ is defined for small $t$ and the curve $t \to F^t (p)$ is the integral curve for X that goes through $p$ at $t = 0$. The Lie derivative of a tensor in the direction of $X$ is defined as the first order term in a suitable Taylor expansion of the tensor when it is moved by the flow of $X$."

I am uncertain of what he means when he refers to the "Taylor expansion of the tensor as it is moved by the flow of $X$". I am only aware of Taylor expansions for functions and I was hoping that somebody can help clarify this.


1 Answer 1


Given a smoothly varying $1$-parameter family $\Psi[t]$ of tensor fields, which we can regard as a map $\Psi: J \to \Gamma(\bigotimes^l TM \otimes \bigotimes^k T^*M)$ for some interval $J \ni 0$, we can differentiate $\Psi$ with respect to $t$ and evaluate at $t = 0$ to produce another such family, $$\partial_t \Psi[t] = \lim_{h \to 0} \frac{\Psi[t + h] - \Psi[t]}{h} \in \Gamma({\textstyle \bigotimes^l TM \otimes \bigotimes^k T^*M}).$$ (With respect to any local coordinates, $\Psi[t]$ has some components $\hat{\Psi}[t]^{b_1 \cdots b_l}{}_{a_1 \cdots a_k}$, and the components of $\partial_t \Psi$ are just the usual single-variable derivatives of these with respect to $t$, i.e., $$\widehat{\partial_t \Psi[t]}^{b_1 \cdots b_l}{}_{a_1 \cdots a_k} = \partial_t \hat{\Psi}[t]^{b_1 \cdots b_l}{}_{a_1 \cdots a_k}. )$$

Now, we can just as easily compute higher derivatives $\partial_t^k \Psi[t]$. Then, just as in the familiar case of Taylor series of functions, we can evaluate all of these derivatives at $t = 0$ and assemble the result into a (formal) Taylor series $$\sum_{k = 0}^{\infty} \frac{1}{k!} \partial^k \Psi[t] \vert_{t = 0}.$$ The (coefficient of) the first-order term here is the tensor field $$\phantom{(\ast)} \qquad \partial_t \Psi[t]\vert_{t = 0} = \left.\lim_{h \to 0} \frac{\Psi[h] - \Psi[0]}{h}\right\vert_{t = 0}. \qquad (\ast)$$

Now, given a tensor field $\Phi \in \Gamma(\bigotimes^l TM \otimes \bigotimes^k T^*M)$ and a vector field $X \in \Gamma(TM)$, the flow $\theta_t$ of $X$ gives us (up to the usual issues involved in existence of flows) a $1$-parameter family of tensor fields $$\Phi[t] := \theta_t^* \Phi$$ with $\Phi[0] = \Phi$. Then, substituting $\Phi[t]$ into the formula $(\ast)$ for the first-order coefficient of the series recovers the usual definition of Lie derivative of a tensor field as claim.

  • $\begingroup$ Thank you for another detailed response. $\endgroup$
    – Memeozuki
    May 31, 2015 at 4:20
  • $\begingroup$ You're welcome, I hope you found it useful. $\endgroup$ May 31, 2015 at 4:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.