LOCAL RESTRICTIONS ON NONPOSITIVELY CURVED n-MANIFOLDS IN Rn+p

In pointwise differential geometry, i.e., linear algebra, we prove two theorems about the curvature operator of isomet-rically immersed submanifolds. We restrict our attention to Euclidean immersions because here the results are most eas-ily 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 1970 result of Weinstein to show that for n-manifolds in Rn+2 the conditions of positive, nonnegative, nonpositive, and negative sectional curvature imply that the curvature operator is positive definite, posi-tive 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 curva-ture at a point x by the rank of the curvature operator at x and prove that surprisingly the rank of a negative semidefi-nite curvature operator is bounded as a function of only the codimension. Specifically, for an n-manifold in Rn+p this rank is at mos