Liouville's equation
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- For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).
In differential geometry, Liouville's equation, named after Joseph Liouville, is the equation satisfied by the conformal factor f of a metric f2(dx2 + dy2) on a surface of constant Gaussian curvature K:
where Δ0 is the flat Laplace operator.
Liouville's equation typically appears in differential geometry books under the heading isothermal coordinates. This term refers to the coordinates x,y, while f can be described as the conformal factor with respect to the flat metric (sometimes the square f2 is referred to as the conformal factor, instead of f itself).
Replacing f by 
, we obtain another commonly found form of the same equation:
- Δ0u = − Ke2u.
[edit] Laplace-Beltrami operator
In a more invariant fashion, the equation can be written in terms of the intrinsic Laplace-Beltrami operator
as follows:
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