Local restrictions on nonpositively curved \(n\)-manifolds in \({\mathbb{R}}^{n+p}\) (Q5929040)

In pointwise differential geometry, i.e., linear algebra, we prove two theorems about the curvature operator of isometrically immersed submanifolds. We restrict our attention to Euclidean immersions because here the results are most easily stated and the curvature operator can be simply expressed as the sum of wedges of second fundamental form matrices. First, we reprove and extend a result of \textit{A. Weinstein} [J. Differ. Geom. 4, 1-4 (1970; Zbl 0194.52903)] to show that for \(n\)-manifolds in \({\mathbb R}^{n+2}\) the conditions of positive, nonnegative, nonpositive, and negative sectional curvature imply that the curvature operator is positive definite, positive semidefinite, negative semidefinite, and negative definite, respectively. We provide a simple example to show that this equivalence is no longer true even in codimension 3. Second, we introduce the concept of measuring the amount of curvature at a point \(x\) by the rank of the curvature operator at \(x\) and prove that surprisingly the rank of a negative semidefinite curvature operator is bounded as a function of only the codimension. Specifically, for an \(n\)-manifold in \({\mathbb R}^{n+p}\) this rank is at most \({p+1\choose 2}\), and this bound is sharp. Under the weaker assumption of nonpositive sectional curvature we prove the rank is at most \(p^3+p^2-p\), and by the proof of the previous theorem this bound can be sharpened to \({p+1\choose 2}\) for \(p = 1\) and \(2\).