How to compute that $\mathcal{L}_Vg_{ij}=g_{ik}\nabla_jV^k+g_{jk}\nabla_iV^k$

Question

How to compute that $\mathcal{L}_Vg_{ij}=g_{ik}\nabla_jV^k+g_{jk}\nabla_iV^k$ ?

The $g_{ij}$ is Riemannian metric, $V=V^k\frac{\partial}{\partial x^k}$ is vector flied ,$\mathcal{L}_Vg$ is Lie derivative.

In fact ,according to the Wiki, I think the $\mathcal{L}_Vg_{ij}=V^k\nabla_k {g_{ij}} +g_{ik}\nabla_jV^k+g_{jk}\nabla_iV^k$.But it's not on book I'm reading.

I also try to compute it according to $\mathcal{L}_X\omega=i_Xd\omega+d(i_X\omega)$, but it's seem to be too complex.

What should compute the Lie derivative? May I have some detail example.

So thanks.

Answer 1

Hint: Consider $g=g^{st}\partial_s\otimes \partial_t$ the use that $\cal L$ is Leibnizian i.e. $${\cal L}_Vg={\cal L}_V(g^{st}\partial_s\otimes \partial_t),$$ $$={\cal L}_V(g^{st})\partial_s\otimes \partial_t+g^{st}{\cal L}_V(\partial_s)\otimes \partial_t+ g^{st}\partial_s\otimes {\cal L}_V(\partial_t),$$ $$={\cal L}_V(g^{st})\partial_s\otimes \partial_t+g^{st}{\cal L}_V(\partial_s)\otimes \partial_t+ g^{st}\partial_s\otimes {\cal L}_V(\partial_t).$$