The fundamental collective degree of freedom of fractional quantum Hall states is identified as a unimodular two-dimensional spatial metric that characterizes the local shape of the correlations of the incompressible fluid. Its quantum fluctuations are controlled by a topologically quantized “guiding-center spin.” Charge fluctuations are proportional to its Gaussian curvature.

• Received 15 June 2011

DOI:https://doi.org/10.1103/PhysRevLett.107.116801