Transversality

Abstract

Let us consider in ℝ² two smoothly embedded circles C ₁, C ₂ (i.e. C ₁ and C ₂ are manifolds in ℝ² which are diffeomorphic to the circle S ¹, cf. Fig. 7.1.1.a). C ₁ and C ₂ are called transversal at an intersection point $\bar x$ if at this point they intersect under an angle unequal to 0 or π; more formally, this means: the tangent spaces of C ₁, C ₂ at $\bar x$ span the whole ℝ² (= space of embedding), cf. Fig. 7.1.1.b. In case of nontransversal intersection, the tangent spaces of C _l and C ₂ at the intersection point coincide, thus spanning a one-dimensional subspace of ℝ² (Fig. 7.1.1.c).