How to prove ##V_{ai;j}=V_{aj;i}## in curved space using the given equation?

[user1139]

Homework Statement

See below.

Relevant Equations

See below.

Question $1$.

Consider the following identity

\begin{equation}
\epsilon^{ij}_{\phantom{ij}k}\epsilon_{i}^{\phantom{i}lm}=h^{jl}h^{m}_{\phantom{m}k}-h^{jm}h^{l}_{\phantom{l}k}
\end{equation}

which we know holds in flat space. Does this identity still hold in curved space? and if so, how does one go about proving it?

Question $2$.

Consider the following

\begin{equation}
\epsilon^{ij}_{\phantom{ij}k}\left(V_{ai;j}-V_{aj;i}\right)=0.
\end{equation}

As $\epsilon^{ij}_{\phantom{ij}k}$ is not arbitrary, one cannot simply conclude that $V_{ai;j}=V_{aj;i}$. Yet, I want to show that one can get $V_{ai;j}=V_{aj;i}$ from the above equation. Is there a way to do that rather than just saying that $V_{ai;j}=V_{aj;i}$ is a possible solution?

 

[Orodruin]

Please define your notation. What are $\epsilon$ and $h$?

 

[user1139]

$\epsilon_{ijk}$ is the Levi-Civita tensor and $h_{ij}$ is the metric tensor in 3-space.

 

[Orodruin]

Are you referring to the object with componens $\pm 1$ an 0 (which is not a tensor but a tensor density - and if so what do you mean by raising/lowering its indices) or to the actual tensor with components $\pm 1$ and 0 in an orthonormal coordinate system?