# Indices of the Riemann Tensor of the first kind

When establishing the identity $$V^i_{,kl}-V^i_{,lk}=-R^i_{tkl}V^t$$ (, denotes covariant differentiation), one of the steps involves raising one of the indices of the Riemann Tensor of the first kind . i.e. $$(g^{ir}R_{trkl})V^t$$ To give $$-R^i_{tkl}V^t$$ However, I expected the result just to be $$R^i_{tkl}V^t$$ I understand that this is incorrect (and there could involve the symmetry properties of the Riemann Tensor).

The question is when raising or lowering indices of an arbitrary tensor of order higher than two using the metric tensor, is there a set of rules for the positions of the raised and lowered indices that must followed?

E.g. if $$R^i_{jkl}=g^{ir}R_{jkrl}=g^{ir}R_{rjkl}=g^{ir}R_{jklr}$$ is true, then it will imply $$R_{jkrl}=R_{rjkl}=R_{jklr}$$, which violates against the symmetry properties of the Riemann Tensor?

## 2 Answers

So the metric is used as the canonical bijection between vectors and covectors ($$v_a = g_{ab} v^b, v^a = g^{ab} v_b$$) in these sorts of expressions and therefore the convention which makes sense is to just state that a given expression is followed by a list of N indices which each must be raised or lowered independently; for example like the symbol, $$T_a{}^b{}_{cd}{}^{e}~.$$This is a variant of a multilinear map taking 5 vector fields to a scalar field (a 5-tensor) which has been adapted with the metric tensor into taking the second and fifth of its arguments as covector fields rather than vector fields; i.e. it is a systematic notation for $$T_a{}^b{}_{cd}{}^e =T_{ab'cde'}~ g^{b'b} ~g^{e'e} ~.$$

In some very limited specific cases we have a convention for which index is raised and we can omit patches of whitespace to form something like $$T^{be}_{acd}.$$ This especially happens if there is some symmetry between the indexes, so for example for the Christoffel symbols since $$\Gamma_{kij} = \Gamma_{kji} = \Gamma_{k[ij]}$$ using antisymmetry brackets ($$A_{[ij]}= (A_{ij} - A_{ji})/2$$), we seldom have any ambiguity if we see $$\Gamma^k_{ij}$$ as the two indices that you would want to remain paired together are the antisymmetric ones. Similarly it is just convention that the Riemann curvature tensor, when it has an upper index, is having the first one as its upper index, even though it is not generally indicated by the symmetry properties, which include $$R_{abcd} = R_{[ab][cd]}.$$

So we would say $$R^a_{bcd} = R^a{}_{bcd}$$more formally.

In this case therefore, $$g^{mb}~R_{abcd} = -g^{mb}R_{bacd} = -R^m_{acd},$$ using both the standard convention for which index is the upper one and the antisymmetry of the first two indexes.

When you raise or lower an index, by contracting with the metric, you must keep the index in the same position. For example,

$$g^{ir}R_{jkrl}=R_{jk}{}^i{}_l.$$

• LaTex gurus: How do I keep the $l$ on the same line as the $jk$? – G. Smith Jun 7 at 16:24
• fixed that for ya. mind, that solution doesn't scale properly as the original symbol scales so it doesn't do right after a \right ] or so. but for this it is sufficient – CR Drost Jun 7 at 16:29
• @CRDrost Thanks! – G. Smith Jun 7 at 22:25