Abstract. The theorem of J. Moser that any two volume elements of equal total volume on a compact manifold are diffeomorphism-equivalent is extended to noncompact manifolds: A necessary and sufficient condition (equal total and same end behavior) is given for diffeomorphism equivalence of two volume forms on a noncompact manifold. Results on the existence of embeddings and immersions with the property of inducing a given volume form are also given. Generalizations to nonorientable manifolds and manifolds with boundary are discussed.The topics of this paper are the action of the diffeomorphism group of a noncompact paracompact oriented manifold on the space of C00 volume forms on the manifold and the existence of volume-form-preserving embeddings of such manifolds into euclidean spaces. The results are essentially generalizations to the case of noncompact manifolds of a theorem of Moser [6] and a corollary of that theorem. The theorem is that if M is a compact connected oriented manifold and if w and t are C°° volume forms on M such that JMu> = Jmt then there is a C°° diffeomorphism M such that tp*r = w. The corollary is that if : M -> M is a diffeomorphism such that
Abstract. Making use of 1-forms and geometric inequalities we prove the rigidity property of complete submanifolds M n with parallel mean curvature normal in a complete and simply connected Riemannian (n-\-p) -manifold N n+P with positive sectional curvature. For given integers n, p and for a nonnegative constant H we find a positive number τ(n,p) 6 (0,1) with the property that if the sectional curvature of TV is pinched in [τ(n,p), 1], and if the squared norm of the second fundamental form is in a certain interval, then N n+P is isometric to the standard unit (n + p)-sphere. As a consequence, such an M is congruent to one of the five models as seen in our Main Theorem.
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