I am looking at a Kahler metric $g$ on a certain manifold $M$, which has the good taste to be invariant under a transitive group of isometries, and I want to say something about its holomorphic sectional curvature.


Now, I can calculate the curvature tensor $R$ of $g$ explicitly at the center of a holomorphic coordinate system. If $(X\_1, \ldots, X\_n)$ is an orthonormal frame of tangent vectors at a given point, then the entries of the tensor are of the form


$$ R\_{j \overline k l \overline m} := R(X\_j, \overline X\_k, X\_l, \overline X\_m) = \text{term 1} \cdot \text{term 2} - \text{term 3} $$


where the terms are sums of Kronecker deltas in $j$, $k$, $l$ and $m$. So if $X = \sum\_j a\_j X\_j$ is a holomorphic tangent vector, which we may assume is of unit norm, the holomorphic sectional curvature in the direction of $X$ is


$$ H(X) = \sum\_{j,k,l,m} a\_j \overline a\_k a\_l \overline a\_m R\_{j \overline k l \overline m}. $$


This is where the pain begins. So far I haven't been able to make any sense of the herd of Kronecker deltas which comes out (except in exceptional cases, like for $R\_{j \overline j k \overline k}$), and I'd be quite happy if I could just let a computer work the damn thing out for me.


**Question:** Is there software that calculates this sort of thing? Or is there a language particularily well adapted to hacking out a script that will calculate this?



Software for calculating products and sums of Kronecker deltas
Yes, there are a number of packages, check out for example:


<http://www.math.washington.edu/~lee/Ricci/>


and more generally


<http://en.wikipedia.org/wiki/Tensor_software>


(I know people use the mathematica packages heavily, and have for the last 25 years, not sure about the other ones).


