We describe a dramatic decrease of the critical velocity in elongated cylindrical Bose-Einstein condensates which originates from the non-uniform character of the radial density profile. We discuss this mechanism with respect to recent measurements at MIT.Superfluidity is one of the striking manifestations of quantum statistics, and the occurence of this phenomenon depends on the excitation spectrum of the quantum liquid. The key physical quantity is the critical velocity v c , that is, the maximum velocity at which the flow of the liquid is still non-dissipative (superfluid). The well-known Landau criterion [1] gives the critical velocity as the minimum ratio of energy to momentum in the excitation spectrum:(we puth = 1 and the particle mass M = 1 [4][5][6] offers new possibilities for the investigation of superfluidity [7]. In the spatially homogeneous case, the spectrum of elementary excitations of a Bosecondensed gas is given by the Bogolyubov dispersion law [8]:and the Landau critical velocity is equal to the speed of sound c s = √ gn 0 (n 0 is the condensate density, g = 4πa, and a > 0 is the scattering length). The first experimental observation of the critical velocity in trapped gaseous condensates, recently reported by the MIT group [9], gives a significantly smaller value of v c .The analyses in [9] and in recent theoretical publications (see e.g. [10,11] and references therein) employ the Feynman hypothesis and provide a qualitative explanation of the MIT experimental result. In this Letter we point out a simple geometrical effect which is characteristic for elongated cylindrical traps. Due to the nonuniform character of the radial density profile, the spectrum of axially propagating excitations in these traps is very different from the Bogolyubov dispersion law (2), and this difference leads to a strong decrease of the critical velocity. We show that this effect can at least partially explain the small critical velocity measured in the MIT experiment [9].We consider an infinitely long cylindrical condensate which is harmonically trapped in the radial (ρ) direction. Then the condensate wave function ψ 0 (ρ) satisfies the Gross-Pitaevskii equationwhere µ is the chemical potential, V (ρ) = ω 2 ρ 2 /2 is the trapping potential, and ω the trap frequency. In the Thomas-Fermi regime, where the ratio η = µ/ω ≫ 1, the density profile is given by n 0 (ρ) ≡ |ψ 0 | 2 = (µ − V (ρ))/g and the chemical potential is related to the maximum condensate density as µ = n 0max g. where φ is the angle in the x, y plane, and the radial functions u mk (ρ) and v mk (ρ) are solutions of the Bogolyubovde Gennes equations (see e.g. [12])Equations (3) and (4) constitute an eigenvalue problem. For given m and k, they lead to a set of frequencies ǫ nm (k) characterized by the radial quantum number n which takes integer values from zero to infinity. In the limit ǫ nm (k) ≪ µ, these modes were found for m = 0 in the hydrodynamic approach in [13][14][15]. For kR ≪ 1, where R = (2µ/ω 2 ) 1/2 is the Thomas-Fermi radial size of 1