Software for calculating products and sums of Kronecker deltas

2011-11-17T10:14:50Z

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?

Answer by Igor Rivin for Software for calculating products and sums of Kronecker deltas

2011-11-17T10:49:52Z

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).

Software for calculating products and sums of Kronecker deltas - MathOverflow

Software for calculating products and sums of Kronecker deltas

Answer by Igor Rivin for Software for calculating products and sums of Kronecker deltas