In mathematics, the Whitney umbrella (or Whitney's umbrella and sometimes called a Cayley umbrella) is a self-intersecting surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line, parallel to the axis of the parabola and lying on its perpendicular bisecting plane.
This formula also includes the negative z axis (which is called the handle of the umbrella).
Whitney's umbrella is a ruled surface and a right conoid. It is important in the field of singularity theory, as a simple local model of a pinch point singularity. The pinch point and the fold singularity are the only stable local singularities of maps from R2 to R3.
In string theory, a Whitney brane is a D7-brane wrapping a variety whose singularities are locally modeled by the Whitney Umbrella. Whitney branes appear naturally when taking Sen's weak coupling limit of F-theory.