# Using symmetry of Riemann tensor to vanish components

The Riemann tensor is skew-symmetric in its first and last pair of indices, i.e.,

\begin{align} R_{abcd} = -R_{abdc} = -R_{bacd} \end{align}

Can I simply use this to say that, for example, the component $$R_{0001} = 0$$ because $$R_{0001} = -R_{0001}$$?

Sometimes it is useful to think of this tensor as a $$6\times6$$ symmetric matrix where the “indices” are $$01$$, $$02$$, $$03$$, $$12$$, $$13$$, and $$23$$.
However, don’t conclude from this that there are $$6+5+4+3+2+1=21$$ independent components. There are actually only $$20$$ because of the algebraic Bianchi identity.
Note that without any of these relations between components, there would be $$4^4=256$$ components! So the Riemann curvature tensor is about 13 times less complicated than it might appear.
• The algebraic Bianchi identity is $R_{abcd}+R_{acdb}+R_{adbc}=0$. Take $abcd=0123$ and you will get a relationship between 3 of the 21 components of the matrix I talked about. If you take another permutation of $0123$ you should get an equivalent relation. If you don’t take $abcd$ to all be different, the identity will reduce to one of the other symmetry relations. Commented Jun 22, 2019 at 21:58