# Curvature of particular Riemannian metric

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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I imagine that you might want to verify the hypothesis of Theorem of Hadamard. You want an explicit or an estimate of the sectional curvature is enough? –  Elias Nov 21 '12 at 13:04
I'm actually more interested in explicit expressions, or good estimates, for the sectional curvature. I'm on a counterexample hunt and will eventually restrict the domain and perturb the metric in the hope of getting something that doesn't have nonpositive sectional curvature, so it's more important for me to be able to calculate or estimate directly what happens then (this case being more of a warmup). –  Gunnar Þór Magnússon Nov 21 '12 at 13:09
In case it helps, this is a disguised version of a metric on the Kahler cone of a product of $n$ projective spaces. There's such a metric on the Kahler cone of any compact manifold. I'm playing around with different types of varieties and trying to break the seminegativity of the sectional curvature of the metric. –  Gunnar Þór Magnússon Nov 21 '12 at 13:14
Suggestion 2 Have you looked in the book of Michael Spivak? A Comprehensive Introduction to Differential Geometry vol. 2 –  Elias Nov 21 '12 at 13:44
@Elias: That's a long book. Is that a general suggestion or did you have a particular passage in mind? –  Gunnar Þór Magnússon Nov 21 '12 at 13:47