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The paper contains a vanishing theorem for P harmonic forms on complete rotationally symmetric Riemannian manifolds. This theorem requires no assumptions on curvature. This paper gives necessary and sufficient conditions for existence of L2 harmonic forms on a special class of Riemannian manifolds. Manifolds of this class were called models by Greene and Wu and played a crucial part in the study of function theory on open manifolds [GW]. Throughout the paper M will denote a model of dimension n, i.e. a C X Riemannian manifold such that: (1) there exists a point o EE M for which the exponential mapping is a diffeomorphism of ToM onto M; (2) every linear isometry p: T,oM > To,M is realized as the differential of an isometry 1: M -M, i.e., ?(o) = o and 0.(o) = p. Clearly, M is complete and can be identified with To,M via exp0. In terms of geodesic polar coordinates (r, 9) E (0, oo) x S'~1 M\{o} the Riemannian metric dj2 of M can be written as S2= dr2 + f(r)2 dO2, (3) where dO2 denotes the standard metric on S'1 and the function f(r) is C on [0, oo) and satisfies f(0) = 0, f'(0) = 1, f(r) > 0 for r > 0 (4) (cf. [S, pp. 179-183]). Complete description of the spaces 9C*(M) of L2 harmonic forms is contained in the following THEOREM. Let M be a model of dimension n > 2. Then (i) 9C(M) = {0} for q # 0, n/2, n, [(0) if f(r) dr = , 0(ii) aV(M) ==d 9M, IR if If(r) l r < X0 Received by the editors July 27, 1978 and, in revised form, December 4, 1978. AMS (MOS) subject classifications (1970). Primary 58G99.

𝐿² harmonic forms on rotationally symmetric Riemannian manifolds