There is an angular bracket operation in geometry, which looks like
$$\langle X,Y\rangle$$
where $X$ and $Y$ are apparently $(0,1)$ tensors. It appears for instance in the answer to the following question:
Riemann, Ricci curvature tensor and Ricci scalar of the n dimensional sphere
Or in the Symmetries and identities section of the following article:
http://en.wikipedia.org/wiki/Riemann_curvature_tensor
Please, let me know what the precise meaning/definition for this bracket operation is (preferably in terms of covariant and contravariant vectors in some coordinate system in components). Thank you!
EDIT:
Mike Miller suggested in the comments that it is the inner product $g(X,Y)$, but if it is just the inner product, then what exactly are $X$ and $Y$ supposed to be in $\text{ric}(X,Y)=(d-1)\langle X,Y\rangle$ in the answer inside the first link above? Do they stand for $X=\sum_{i=0}^{d-1}dx_i$ and $Y$ similar?
Or maybe, if $g(\cdot,\cdot)=\sum_{i,j}g_{ij} \, dx_i \, dx_j$ then rather $X=\sum_{i=0}^{d-1}\partial_i$ and $Y$ similar?
$\langle X,Y\rangle$ is another notation for $g(X,Y)$. The answer given is showing that for arbitrary vector fields on $S^n$, $X,Y$, we have $\text{ric}(X,Y)=(n-1)g(X,Y)$; that is, that $\text{ric}=(n-1)g$. So the sphere is an Einstein manifold with constant $k=n-1$.