conformal_change_formulae
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| conformal_change_formulae [2022/08/31 13:43] – created spencer | conformal_change_formulae [2022/08/31 13:46] (current) – spencer | ||
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| + | ====== Conformal change formulae ====== | ||
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| Suppose throughout that $M^n$ is a manifold with two conformal Riemannian metrics $\widetilde{g} = e^{2\phi}g$. | Suppose throughout that $M^n$ is a manifold with two conformal Riemannian metrics $\widetilde{g} = e^{2\phi}g$. | ||
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| Thus at $p$, | Thus at $p$, | ||
| \begin{align*} | \begin{align*} | ||
| - | \mathrm{vol}(\widetilde{g}) &= \sqrt{(e^{2\phi})^n} \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n \\ | + | \mathrm{vol}(\widetilde{g}) |
| + | &= \sqrt{(e^{2\phi})^n} | ||
| &= e^{\phi n} \mathrm{vol}(g). | &= e^{\phi n} \mathrm{vol}(g). | ||
| \end{align*} | \end{align*} | ||
conformal_change_formulae.1661967814.txt.gz · Last modified: by spencer