AbstractLetXn⊂orXn⊂ ℙn + abe a patch of aC∞submanifold of an affine or projective space such that through each pointx∈Xthere exists a line osculating to ordern+ 1 atx. We show thatXis uniruled by lines, generalizing a classical theorem for surfaces. We describe two circumstances that imply linear spaces of dimensionkosculating to order two must be contained inX, shedding light on some of Ein's results on dual varieties. We present some partial results on the general problem of finding the integerm0=m0(k, n, a) such that there exist examples of patchesXn⊂ ℙn+a, having a linear spaceLof dimensionkosculating to orderm0— 1 at each point such thatLis not locally contained inX, but if there arek-planes osculating to orderm0at each point, they are locally contained inX.The same conclusions hold in the analytic category and complex analytic category if there is a linear space osculating to ordermat one general pointx∈X.