# Some questions on tensors

user1139
Homework Statement:
See below.
Relevant Equations:
See below.
Question ##1##.

Consider the following identity

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

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

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

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?

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user1139
##\epsilon_{ijk}## is the Levi-Civita tensor and ##h_{ij}## is the metric tensor in 3-space.

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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?