Does the order of the metric and tensor matter when raising or lowering indices?

[binbagsss]

This is probably a really stupid question , but,
Does it matter whether the metric is after or before the tensor?

My guess is it doesn't because tensors can be positioned in any order, the equation is unchanged.

E.g $M_{ab}B^{c}T^{m}_{nl}=T^{m}_{nl}M_{ab}B^{c}$ right?

However the covariant derivative operates on tensor after it and not the tensors that are before it,

So

$g_{uv}V^{v}=V_{u}$

what about $V^{v}g_{uv}$ does this equal $V_{u}$ or not,

Many thanks

 

[Orodruin]

Does it matter whether the metric is after or before the tensor?

No.