I am trying solve a question from the text Mathematical Analysis: An Introduction by Andrew Browder. On page 266 in chapter 11 on Manifolds, question 10, it asks
Let $\ M\ $and$\ N\ $be 2-manifolds in$\ {\mathbb{R}}^{3}. \ $ Show that if the tangent spaces to $\ M\ $and$\ N\ $do not conincide at any point of$\ M\cap\ N,\ $then$\ M\cap\ N\ $is a 1-manifold, and the tangent line to$\ M\cap\ N\ $at $\ p\in\ M\cap\ N\ $is the intersection of the tangent planes to$\ M\ $and$\ N\ $at$\ p.\ $
I am having trouble with the wording of the premise of the question. What does it mean for tangent spaces to manifolds M and N to not intersect? Does it mean that the set of vectors that form the basis of null spaces derivative matrix for M and N respectively consist two different set of basis vectors. Intuitively the intersection of two 2-manifolds automatically make it either a curve or a set of two different curves.