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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-Weitzenbock formulas for gradients. As applications, we give some vanishing theorems.

Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds