Geometrical approach to bosonization ofD>1 dimensional (non)-Fermi liquids

We discuss an approach to higher-dimensional bosonization of interacting fermions based on a picture of fluctuating Fermi surface. Compared with the linearized ``constructive'' approach this method allows an account of the Fermi surface curvature due to non-Gaussian terms in the bosonized Lagrangian. On the basis of this description we propose a procedure of calculating density response functions beyond the random-phase approximation (RPA). We also formulate a bosonic theory of the compressible metal-like state at the half-filled lowest Landau level and check that in the Gaussian approximation it reproduces RPA results of the gauge theory by Halperin, Lee, and Read.