According the first line on page $2$ of this paper,
A smooth vector field $\xi$ on a Riemannian manifold $(M, g)$ is said to be a conformal vector field if its flow consists of conformal transformations or, equivalently, if there exists a smooth function $f$ on $M$ (called the potential function of the conformal vector field $\xi$) that satisfies $\mathscr{L}_{\xi} g = 2fg$, where $\mathscr{L}_{\xi} g$ is the Lie derivative of $g$ with respect $\xi$.
By the other hand, this paper that I'm reading define in a different way:
A vector field $X$ is conformal if $\nabla_j X_i + \nabla_i X_j = 2 \lambda g_{ij}$ for a function $\lambda$.
I would like to know how can I see the Lie derivative of the tensor metric $g$ in terms of Levi-Civitta connection.
I'm not familiar with the Lie's derivative, then I saw in Lee's Introduction to Smooth Manifold the following corollary:
$\textbf{Corollary 12.33.}$ If $V$ is a smooth vector field and $A$ is a smooth covariant $k$-tensor field, then for any smooth vector fields $X_1, \cdots, X_k$ ,
$$\mathscr{L}_V A = V(A(X_1, \cdots, X_k)) – A([V,X_1], X_2, \cdots, X_k) – \cdots – A(X_1, \cdots, X_{k-1}, [V, X_k]).$$
Denoting by $\partial_i := \frac{\partial}{\partial x_i}$, defining $X = X^k \partial_k$ and applying this corollary to the tensor metric, I obtained
\begin{align*}
\mathscr{L}_X g &= X(g(\partial_i,\partial_j)) – g([X,\partial_i], \partial_j) – g(\partial_i, [X,\partial_j])\\
&= X^k \frac{\partial g_{ij}}{\partial x_k} + g \left( \frac{\partial X^k}{\partial x_i} \partial_i, \partial_j \right) + g \left( \partial_i, \frac{\partial X^k}{\partial x_j} \partial_j \right)\\
&= X^k \frac{\partial g_{ij}}{\partial x_k} + \frac{\partial X^k}{\partial x_i} g_{ij} + \frac{\partial X^k}{\partial x_j} g_{ij}.
\end{align*}
I'm stuck here.
I also read on this Wikipedia's article that
$\mathscr{L}_X g = (X^c g_{ab \ ; \ c} + g_{cb} X_{; \ a}^c + g_{ac} X_{; \ b}^c ) dx^a \otimes dx^b = (X_{b \ ; a} + X_{a \ ; b})dx^a \otimes dx^b$. (This is the last example of the section of Coordinate expressions and was explained in the beginning of this section the notation "$;$")
I didn't understand how this computation was done, but it seems that the notation "$;$" is the same of "$\nabla$" given in the second paper linked, which lead me to think that $\nabla_i X^j$ it's just a notation for the covariant derivative of a coordinate $X^j$ of the vector field $X^k \partial x_k$ in the direction $\partial x_i$, if I'm right, then the work it's just understand why $\mathscr{L}_X g = (X_{b \ ; a} + X_{a \ ; b})dx^a \otimes dx^b$. Am I right? If I'm right, then how can I deduce the expression above?
Thanks in advance!
Best Answer
I found on the final of the page $14$ of this thesis how to prove that $\nabla_i X_j + \nabla_j X_i = (\mathscr{L}_X g)_{ij}$, which is more clear than in Wald's book. I will put here the development: