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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?

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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).

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How delightfully timesaving. –  Gunnar Magnusson Nov 17 '11 at 12:05
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