Is a linear space contained in a submanifold? On the number of derivativesneeded to tell

Let X-n subset of A(n) (+) (a) or X-n subset of P-n (+) (a) be a patch of a C-infinity submanifold of an affine or projective space such that through each point x is an element of X there exists a line osculating to order n 1 at x. We show that X is uniruled by lines, generalizing a classical theor em for surfaces. We describe two circumstances that imply linear spaces of dimension k osculating to order two must be contained in X, shedding light on some of fin's results on dual varieties. We present some partial results on the general problem of finding the integer m(0) = m(0) (k, n, a) such t hat there exist examples of patches X-n subset of P-n (+) (a), having a lin ear space L of dimension k osculating to order m(0)-1 at each point such th at L is not locally contained in X, but if there are k-planes osculating to order m(0), at each point, they are locally contained in X.