Is a linear space contained in a submanifold? – On the number of derivatives needed to tell

Let Xn ⊂ or Xn ⊂ ℙn + a be a patch of a C∞ submanifold of an affine or projective space such that through each point x ∈ X there exists a line osculating to order n + 1 at x. We show that X is uniruled by lines, generalizing a classical theorem 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 Ein's results on dual varieties. We present some partial results on the general problem of finding the integer m0 = m0(k, n, a) such that there exist examples of patches Xn ⊂ ℙn + a, having a linear space L of dimension k osculating to order m0 — 1 at each point such that L is not locally contained in X, but if there are k-planes osculating to order m0 at each point, they are locally contained in X. The same conclusions hold in the analytic category and complex analytic category if there is a linear space osculating to order m at one general point x ∈ X.