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Let $U = \{ (x_1, \dots, x_n) \mid x_j > 0 \text{ for all } j\}$ and let $\|x\|^2 = \sum_j x_j^2$. The function $x \mapsto -\log \|x\|^2$ is strictly convex on $U$ and thus defines a Riemannian metric $g$ on that space. Up to a factor of $\tfrac 12$ that we don't care about, this metric is $ g = \frac 1{\|x\|^4} \tau - \frac 1{\|x\|^2} g_{st}, $ where $\tau$ is the tensor described by the matrix $\tau = (x_jx_k)_{1 \leq j \leq n, 1 \leq k \leq n}$ in these coordinates, and $g_{st}$ is the standard Euclidean metric.

I want to know the sectional curvature of $g$ (and eventually of metrics that are perturbations of $g$). Does anyone know a nice trick or method to calculate this?

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    Margnusson See the Handbook Modern Differential Geometry of Curves And Surfaces With Mathematica. In Google Book's http://books.google.com.br/books?id=owEj9TMYo7IC&dq=Modern+Differential+Geometry+of+Curves+And+Surfaces+With+Mathematica&hl=pt-BR&sa=X&ei=7N2sUIyiO-e80QG--YGACw&ved=0CDYQ6AEwAA2012-11-21

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