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Expanding on Willie's comment: there is work of Schoen--Yau and Gromov--Lawson on obstructions to the existence of a metric with positive scalar curvature. One consequence is that if a metric on $\mathbb{T}^n$ has nonnegative scalar curvature, then that metric is flat. So, at least in certain cases, "nonnegative scalar curvature" is a rigid property of a metric.
Note that the curvature term in the Weitzenböck formula is only the Ricci curvature when $\omega$ is a $1$-form. In general it involves the Riemannian curvature tensor. However, it always acts on $\omega$ as a zeroth-order operator, so this has no effect on the principal symbol.
