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Abstract. Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

We first consider nonlinear Grassmann sigma models in any dimension and next construct their submodels. For these models we construct an infinite number of nontrivial conserved currents.Our result is independent of time-space dimensions and, therfore, is a full generalization of that of authors (Alvarez, Ferreira and Guillen).Our result also suggests that our method may be applied to other nonlinear sigma models such as chiral models, G/H sigma models in any dimension. IntroductionNonlinear (Grassmann) sigma models in two dimensions are very interesting objects to study in the not only classical but also quantum point of view and we have a great many papers on this topics. See, for example, Zakrzewski

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion Kähler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita-Schwinger fields. In the case of Calabi-Yau, hyperkähler, G 2 and Spin(7) manifolds we find an identification of the kernel of the Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita-Schwinger fields.2000 Mathematics Subject Classification: Primary 32Q20, 57R20, 53C26, 53C27 53C35, 53C15.

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