Software for calculating products and sums of Kronecker deltas - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:12:07Z http://mathoverflow.net/feeds/question/81161 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81161/software-for-calculating-products-and-sums-of-kronecker-deltas Software for calculating products and sums of Kronecker deltas Gunnar Magnusson 2011-11-17T10:14:50Z 2011-11-17T10:49:52Z <p>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.</p> <p>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</p> <p>$$R_{j \overline k l \overline m} := R(X_j, \overline X_k, X_l, \overline X_m) = \text{term 1} \cdot \text{term 2} - \text{term 3}$$</p> <p>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</p> <p>$$H(X) = \sum_{j,k,l,m} a_j \overline a_k a_l \overline a_m R_{j \overline k l \overline m}.$$</p> <p>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.</p> <p><strong>Question:</strong> 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?</p> http://mathoverflow.net/questions/81161/software-for-calculating-products-and-sums-of-kronecker-deltas/81163#81163 Answer by Igor Rivin for Software for calculating products and sums of Kronecker deltas Igor Rivin 2011-11-17T10:49:52Z 2011-11-17T10:49:52Z <p>Yes, there are a number of packages, check out for example:</p> <p><a href="http://www.math.washington.edu/~lee/Ricci/" rel="nofollow">http://www.math.washington.edu/~lee/Ricci/</a></p> <p>and more generally</p> <p><a href="http://en.wikipedia.org/wiki/Tensor_software" rel="nofollow">http://en.wikipedia.org/wiki/Tensor_software</a></p> <p>(I know people use the mathematica packages heavily, and have for the last 25 years, not sure about the other ones).</p>