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Given a semi-Riemannian manifold $(M,g)$; for $p \in M$, we define the sectional curvature of a non-degenerate $2$-plane $\sigma$ with basis $\{u,v\}$ as $$K(\sigma) := K(u,v) = \frac{R(u,v,v,u)}{Q(u,v)}$$ where $R$ is the Riemann-curvature tensor and $$Q(u,v) := g(u,u)g(v,v)-g(u,v)^2.$$

We set $\tilde{u} = au+bv$ and $\tilde{v} = cu+dv$. Could someone guide me through the computation that $$Q(\tilde{u},\tilde{v}) = (ac-bd)^2Q(u,v)?$$

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2 Answers 2

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It's quite simple if you use the fact that the riemannian metric is, in fact, a metric. That is, its bilinearity: \begin{align} Q(\overline{u}, \overline{v}) &= Q(au+bv,cu+dv) \\ &= g(au+bv,au+bv)g(cu+dv,cu+dv) - g(au+bv,cu+dv)^2 \\ &= (a^2g(u,u) + 2abg(u,v) + b^2g(v,v)) \cdot (c^2g(u,u) + 2cdg(u,v) + d^2g(v,v)) \\ &- (acg(u,u) + (bc+ad)g(u,v) + bdg(v,v))^2 \end{align}

Expanding the product and the square will make the right terms cancel out. I can edit the answer and keep up with the calculations if you still don't get the picture.

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  • $\begingroup$ Yeah, I got that far, but then I read in some notes "by a simple calculation we get..." and thought that there was an easier way to get the answer. :) $\endgroup$
    – Ben123
    Aug 2, 2023 at 14:01
  • $\begingroup$ But then I suspect I just have to "soldier through" that calculation. $\endgroup$
    – Ben123
    Aug 2, 2023 at 14:01
  • $\begingroup$ The part about "by a simple calculation" could well have referred to the analogous calculation to show that $$R(\tilde{u},\tilde{v},\tilde{v},\tilde{u}) = (ad-bc)^2R(u,v,v,u).$$ $\endgroup$
    – Ben123
    Aug 7, 2023 at 14:54
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To answer my own question (thanks to Lucas for answering):

$$Q(\tilde{u},\tilde{v}) = Q(au+bv,cu+dv) = \cdots = (a^2g_{u,u}+2abg_{u,v}+b^2g_{v,v}) \cdot (c^2g_{u,u}+2cdg_{u,v}+d^2g_{v,v})-(acg_{u,u}+(bc+ad)g_{u,v}+bdg_{v,v}) = \cdots = ((ad)^2-2abcdg+(bc)^2)g_{u,u}g_{v,v}-((bc+ad)^2-4abcd)g_{u,v}^2 = (ad-bc)^2g_{u,u}g_{v,v}-((ad)^2+(bc)^2+2acbd-4acbd))g_{u,v}^2 = (ad-bc)^2g_{u,u}g_{v,v}-(ad-bc)^2g_{u,v}^2 = (ad-bc)^2(g_{u,u}g_{v,v}-g_{u,v}^2) = (ad-bc)^2Q(u,v)$$

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