📂Geometry

Tangent Points and Transversal Points in Geometry

Terminology

Let’s denote the intersection of two manifolds as $p$. If two manifolds merely brush against each other, this point is called a tangent point, and if they intersect transversely, it is called a transversal point.

Explanation

The reason for explaining tangent and transversal points as ’terms’ without rigorously defining them mathematically is because these concepts are often used intuitively across mathematics without a need for separate definitions.

For instance, in the case of a $1$-dimensional manifold like a curve, it might seem possible to define it using a tangent vector, but actually, you have to add the condition of being differentiable at $p$, which prevents covering all the intended concepts. Geometrically, whether they are tangent or transversal should have no relation to differentiability, and indeed this is a chronic problem faced even when generalized to $n$-dimensional manifolds.

Additionally, it is important to note that the concepts of tangent and transversal points are valid even if the dimensions of the manifolds differ. For example, if a curve pierces a surface perpendicularly, the point where they meet should be a transversal point. To discuss such a scenario, we should not specify the dimensions of the manifolds in the premise.