Local restrictions on nonpositively curved n-manifolds in R-n plus p

In pointwise differential geometry, i.e., linear algebra, we prove two theo rems about the curvature operator of isometrically immersed submanifolds. W e restrict our attention to Euclidean immersions because here the results a re most easily stated and the curvature operator can be simply expressed as the sum of wedges of second fundamental form matrices. First, we reprove a nd extend a 1970 result of Weinstein to show that for n-manifolds in Rn+2 t he 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 a mount 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 o perator is bounded as a function of only the codimension. Specifically, for an n-manifold in Rn+p this rank is at most ((p+1)(2)), and this bound is s harp. Under the weaker assumption of nonpositive sectional curvature we pro ve the rank is at most p(3) + p(2) - p, and by the proof of the previous th eorem this bound can be sharpened to ((p+1)(2)) for p = 1 and 2.